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H|$E1I}AUD!9wLJDIL9|$u[1OfAWAVAUATMUHSHt$H~pHAHHfnLHfpHLHH9uHH@t,t$HHHA4H9~HAtH9~AtH[]A\A]A^A_@IӃEDrE1Dt$\$ H~f.I}AUt$I9v$D$ 19s@I}AUI9wH L$C IL9uH[]A\A]A^A_fIII LHI LHI LHI LHA HE1I}AUD!9rD$CIL9uH[]A\A]A^A_ÐH1DI}AUD$AHH9uH[]A\A]A^A_1zAWAVAUATUSLHfHHAHCHfnLHfaHfpLʐHH9uHHt\Hxf4CHH9~KHxftH9~=HxftH9~/HxftH9~!HxftH9~Hft H9}ft H[]A\A]A^A_DIAfHEH~DrII1H$A1D$ ufI?AWfD9s@D$ Af9s1t1fA9vI?AWȉfA9wۿHDfKH;$2t1뒐AHHH HHH HHH HH HM4I11t@1!fA9sI?AW!fA9rDHfSL9uH[]A\A]A^A_@HI,IHDfCH9mI?AWATfHSH9uP1DAWAVIAUATUSH@t$ uVH~>> from numpy.random import Generator, PCG64 >>> rng = Generator(PCG64()) >>> rng.standard_normal() -0.203 # random See Also -------- default_rng : Recommended constructor for `Generator`. Gets the bit generator instance used by the generator Returns ------- bit_generator : BitGenerator The bit generator instance used by the generator Bad call flags for CyFunction__pyx_capi____loader__loader__file__origin__package__parent__path__submodule_search_locations%.200s() needs an argumentkeywords must be stringsexactly__getstate__Missing type objectcannot import name %Sendunparsable format string'complex double''signed char''unsigned char''short''unsigned short''int''unsigned int''long''unsigned long''long long''unsigned long long''double''complex long double''bool''char''complex float''float'a structPython objecta pointera string'long double'an integer is required%s (%s:%d)View.MemoryView._erris_f_contigis_c_contigView.MemoryView.Enum.__init____reduce__numpy/random/_generator.pyxView.MemoryView._err_dimView.MemoryView._err_extentsat mostat least__cinit__BitGenerator__setstate_cython__View.MemoryView.memview_slicememviewsliceobjtupleExpected %s, got %.200s__reduce_cython____setstate__logisticgumbelspawnrayleighpoisson'NoneType' is not iterableView.MemoryView._unellipsifystandard_cauchywaldvonmisesnoncentral_chisquarebeta__pyx_unpickle_Enumpowerparetologseriesstandard_tzipfweibullname '%U' is not definedpermutationdirichletmultivariate_hypergeometricmultivariate_normalvhutriangularstandard_gammastandard_normaluniformassignmentbuffer dtypeBuffer not C contiguous.choiceintegersrandombuiltinscython_runtime__builtins__does not match_cython_3_0_114294967296complexnumpyflatiterbroadcastndarraygenericnumberunsignedintegerinexactcomplexfloatingflexiblecharacterufuncnumpy.random.bit_generatorSeedSequenceSeedlessSequencenumpy.random._commondoubleLEGACY_POISSON_LAM_MAXuint64_tMAXSIZE_rand_uint64_rand_uint32_rand_uint16_rand_uint8_rand_bool_rand_int64_rand_int32_rand_int16_rand_int8check_constraintcheck_array_constraintdouble (double *, npy_intp)kahan_sumdouble_fillfloat_fillvalidate_output_shapecontdisccont_fcont_broadcast_3discrete_broadcast_iiiView.MemoryViewnumpy._core._multiarray_umathnumpy.core._multiarray_umath_ARRAY_API_ARRAY_API is NULL pointernumpy/__init__.cython-30.pxdnumpy.import_arrayinit numpy.random._generatordefault_rngformatcopy_fortrancopypermutedoutnoncentral_fshufflemultinomialnumpy.PyArray_MultiIterNew2numpy.PyArray_MultiIterNew3negative_binomialstandard_exponentiallaplacelognormal__module__func_doc__doc__func_name__name____qualname__func_dict__dict__func_globals__globals__func_closure__closure__func_code__code__func_defaults__defaults____kwdefaults____annotations___is_coroutineTbasestridessuboffsetsndimitemsizenbytes__repr__numpy.random._generator.Enumnumpy.random._generator.arraymemview__getattr___bit_generatorL,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,d,,4,,,,,,,,,(,,,,,,,,,,,,,,,p,зXn,,z,,,]@,]26666666666666666666666666666666666666666666666666666666666666666>66666bn66z66666666666666666&6д66J666V6|||ڸǸ|#ٴٴ#ٴٴٴٴٴ##ٴٴ#ٴٴ==#ٴٴٴٴٴٴٴٴٴٴٴٴٴٴٴٴ[-ٴ--ٴٴٴٴٴٴɷܷfff{˸źo{0o000000No0000000000ǹ 77)))dwmhm\mPmDmPwmmuuP8x++B(9(0(,yyPvGv>v5v,vyMyvvwjwy 8Hyy|Ǐ?liؔȔpzqh ?|?<<<B90TD<__pyx_fatalerror00010203040506070809101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899default_rng(seed=None) Construct a new Generator with the default BitGenerator (PCG64). Parameters ---------- seed : {None, int, array_like[ints], SeedSequence, BitGenerator, Generator}, optional A seed to initialize the `BitGenerator`. If None, then fresh, unpredictable entropy will be pulled from the OS. If an ``int`` or ``array_like[ints]`` is passed, then all values must be non-negative and will be passed to `SeedSequence` to derive the initial `BitGenerator` state. One may also pass in a `SeedSequence` instance. Additionally, when passed a `BitGenerator`, it will be wrapped by `Generator`. If passed a `Generator`, it will be returned unaltered. Returns ------- Generator The initialized generator object. Notes ----- If ``seed`` is not a `BitGenerator` or a `Generator`, a new `BitGenerator` is instantiated. This function does not manage a default global instance. See :ref:`seeding_and_entropy` for more information about seeding. Examples -------- `default_rng` is the recommended constructor for the random number class `Generator`. Here are several ways we can construct a random number generator using `default_rng` and the `Generator` class. Here we use `default_rng` to generate a random float: >>> import numpy as np >>> rng = np.random.default_rng(12345) >>> print(rng) Generator(PCG64) >>> rfloat = rng.random() >>> rfloat 0.22733602246716966 >>> type(rfloat) Here we use `default_rng` to generate 3 random integers between 0 (inclusive) and 10 (exclusive): >>> import numpy as np >>> rng = np.random.default_rng(12345) >>> rints = rng.integers(low=0, high=10, size=3) >>> rints array([6, 2, 7]) >>> type(rints[0]) Here we specify a seed so that we have reproducible results: >>> import numpy as np >>> rng = np.random.default_rng(seed=42) >>> print(rng) Generator(PCG64) >>> arr1 = rng.random((3, 3)) >>> arr1 array([[0.77395605, 0.43887844, 0.85859792], [0.69736803, 0.09417735, 0.97562235], [0.7611397 , 0.78606431, 0.12811363]]) If we exit and restart our Python interpreter, we'll see that we generate the same random numbers again: >>> import numpy as np >>> rng = np.random.default_rng(seed=42) >>> arr2 = rng.random((3, 3)) >>> arr2 array([[0.77395605, 0.43887844, 0.85859792], [0.69736803, 0.09417735, 0.97562235], [0.7611397 , 0.78606431, 0.12811363]]) permutation(x, axis=0) Randomly permute a sequence, or return a permuted range. Parameters ---------- x : int or array_like If `x` is an integer, randomly permute ``np.arange(x)``. If `x` is an array, make a copy and shuffle the elements randomly. axis : int, optional The axis which `x` is shuffled along. Default is 0. Returns ------- out : ndarray Permuted sequence or array range. Examples -------- >>> rng = np.random.default_rng() >>> rng.permutation(10) array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6]) # random >>> rng.permutation([1, 4, 9, 12, 15]) array([15, 1, 9, 4, 12]) # random >>> arr = np.arange(9).reshape((3, 3)) >>> rng.permutation(arr) array([[6, 7, 8], # random [0, 1, 2], [3, 4, 5]]) >>> rng.permutation("abc") Traceback (most recent call last): ... numpy.exceptions.AxisError: axis 0 is out of bounds for array of dimension 0 >>> arr = np.arange(9).reshape((3, 3)) >>> rng.permutation(arr, axis=1) array([[0, 2, 1], # random [3, 5, 4], [6, 8, 7]]) shuffle(x, axis=0) Modify an array or sequence in-place by shuffling its contents. The order of sub-arrays is changed but their contents remains the same. Parameters ---------- x : ndarray or MutableSequence The array, list or mutable sequence to be shuffled. axis : int, optional The axis which `x` is shuffled along. Default is 0. It is only supported on `ndarray` objects. Returns ------- None See Also -------- permuted permutation Notes ----- An important distinction between methods ``shuffle`` and ``permuted`` is how they both treat the ``axis`` parameter which can be found at :ref:`generator-handling-axis-parameter`. Examples -------- >>> rng = np.random.default_rng() >>> arr = np.arange(10) >>> arr array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]) >>> rng.shuffle(arr) >>> arr array([2, 0, 7, 5, 1, 4, 8, 9, 3, 6]) # random >>> arr = np.arange(9).reshape((3, 3)) >>> arr array([[0, 1, 2], [3, 4, 5], [6, 7, 8]]) >>> rng.shuffle(arr) >>> arr array([[3, 4, 5], # random [6, 7, 8], [0, 1, 2]]) >>> arr = np.arange(9).reshape((3, 3)) >>> arr array([[0, 1, 2], [3, 4, 5], [6, 7, 8]]) >>> rng.shuffle(arr, axis=1) >>> arr array([[2, 0, 1], # random [5, 3, 4], [8, 6, 7]]) permuted(x, axis=None, out=None) Randomly permute `x` along axis `axis`. Unlike `shuffle`, each slice along the given axis is shuffled independently of the others. Parameters ---------- x : array_like, at least one-dimensional Array to be shuffled. axis : int, optional Slices of `x` in this axis are shuffled. Each slice is shuffled independently of the others. If `axis` is None, the flattened array is shuffled. out : ndarray, optional If given, this is the destination of the shuffled array. If `out` is None, a shuffled copy of the array is returned. Returns ------- ndarray If `out` is None, a shuffled copy of `x` is returned. Otherwise, the shuffled array is stored in `out`, and `out` is returned See Also -------- shuffle permutation Notes ----- An important distinction between methods ``shuffle`` and ``permuted`` is how they both treat the ``axis`` parameter which can be found at :ref:`generator-handling-axis-parameter`. Examples -------- Create a `numpy.random.Generator` instance: >>> rng = np.random.default_rng() Create a test array: >>> x = np.arange(24).reshape(3, 8) >>> x array([[ 0, 1, 2, 3, 4, 5, 6, 7], [ 8, 9, 10, 11, 12, 13, 14, 15], [16, 17, 18, 19, 20, 21, 22, 23]]) Shuffle the rows of `x`: >>> y = rng.permuted(x, axis=1) >>> y array([[ 4, 3, 6, 7, 1, 2, 5, 0], # random [15, 10, 14, 9, 12, 11, 8, 13], [17, 16, 20, 21, 18, 22, 23, 19]]) `x` has not been modified: >>> x array([[ 0, 1, 2, 3, 4, 5, 6, 7], [ 8, 9, 10, 11, 12, 13, 14, 15], [16, 17, 18, 19, 20, 21, 22, 23]]) To shuffle the rows of `x` in-place, pass `x` as the `out` parameter: >>> y = rng.permuted(x, axis=1, out=x) >>> x array([[ 3, 0, 4, 7, 1, 6, 2, 5], # random [ 8, 14, 13, 9, 12, 11, 15, 10], [17, 18, 16, 22, 19, 23, 20, 21]]) Note that when the ``out`` parameter is given, the return value is ``out``: >>> y is x True dirichlet(alpha, size=None) Draw samples from the Dirichlet distribution. Draw `size` samples of dimension k from a Dirichlet distribution. A Dirichlet-distributed random variable can be seen as a multivariate generalization of a Beta distribution. The Dirichlet distribution is a conjugate prior of a multinomial distribution in Bayesian inference. Parameters ---------- alpha : sequence of floats, length k Parameter of the distribution (length ``k`` for sample of length ``k``). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n)``, then ``m * n * k`` samples are drawn. Default is None, in which case a vector of length ``k`` is returned. Returns ------- samples : ndarray, The drawn samples, of shape ``(size, k)``. Raises ------ ValueError If any value in ``alpha`` is less than zero Notes ----- The Dirichlet distribution is a distribution over vectors :math:`x` that fulfil the conditions :math:`x_i>0` and :math:`\sum_{i=1}^k x_i = 1`. The probability density function :math:`p` of a Dirichlet-distributed random vector :math:`X` is proportional to .. math:: p(x) \propto \prod_{i=1}^{k}{x^{\alpha_i-1}_i}, where :math:`\alpha` is a vector containing the positive concentration parameters. The method uses the following property for computation: let :math:`Y` be a random vector which has components that follow a standard gamma distribution, then :math:`X = \frac{1}{\sum_{i=1}^k{Y_i}} Y` is Dirichlet-distributed References ---------- .. [1] David McKay, "Information Theory, Inference and Learning Algorithms," chapter 23, https://www.inference.org.uk/mackay/itila/ .. [2] Wikipedia, "Dirichlet distribution", https://en.wikipedia.org/wiki/Dirichlet_distribution Examples -------- Taking an example cited in Wikipedia, this distribution can be used if one wanted to cut strings (each of initial length 1.0) into K pieces with different lengths, where each piece had, on average, a designated average length, but allowing some variation in the relative sizes of the pieces. >>> rng = np.random.default_rng() >>> s = rng.dirichlet((10, 5, 3), 20).transpose() >>> import matplotlib.pyplot as plt >>> plt.barh(range(20), s[0]) >>> plt.barh(range(20), s[1], left=s[0], color='g') >>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r') >>> plt.title("Lengths of Strings") multivariate_hypergeometric(colors, nsample, size=None, method='marginals') Generate variates from a multivariate hypergeometric distribution. The multivariate hypergeometric distribution is a generalization of the hypergeometric distribution. Choose ``nsample`` items at random without replacement from a collection with ``N`` distinct types. ``N`` is the length of ``colors``, and the values in ``colors`` are the number of occurrences of that type in the collection. The total number of items in the collection is ``sum(colors)``. Each random variate generated by this function is a vector of length ``N`` holding the counts of the different types that occurred in the ``nsample`` items. The name ``colors`` comes from a common description of the distribution: it is the probability distribution of the number of marbles of each color selected without replacement from an urn containing marbles of different colors; ``colors[i]`` is the number of marbles in the urn with color ``i``. Parameters ---------- colors : sequence of integers The number of each type of item in the collection from which a sample is drawn. The values in ``colors`` must be nonnegative. To avoid loss of precision in the algorithm, ``sum(colors)`` must be less than ``10**9`` when `method` is "marginals". nsample : int The number of items selected. ``nsample`` must not be greater than ``sum(colors)``. size : int or tuple of ints, optional The number of variates to generate, either an integer or a tuple holding the shape of the array of variates. If the given size is, e.g., ``(k, m)``, then ``k * m`` variates are drawn, where one variate is a vector of length ``len(colors)``, and the return value has shape ``(k, m, len(colors))``. If `size` is an integer, the output has shape ``(size, len(colors))``. Default is None, in which case a single variate is returned as an array with shape ``(len(colors),)``. method : string, optional Specify the algorithm that is used to generate the variates. Must be 'count' or 'marginals' (the default). See the Notes for a description of the methods. Returns ------- variates : ndarray Array of variates drawn from the multivariate hypergeometric distribution. See Also -------- hypergeometric : Draw samples from the (univariate) hypergeometric distribution. Notes ----- The two methods do not return the same sequence of variates. The "count" algorithm is roughly equivalent to the following numpy code:: choices = np.repeat(np.arange(len(colors)), colors) selection = np.random.choice(choices, nsample, replace=False) variate = np.bincount(selection, minlength=len(colors)) The "count" algorithm uses a temporary array of integers with length ``sum(colors)``. The "marginals" algorithm generates a variate by using repeated calls to the univariate hypergeometric sampler. It is roughly equivalent to:: variate = np.zeros(len(colors), dtype=np.int64) # `remaining` is the cumulative sum of `colors` from the last # element to the first; e.g. if `colors` is [3, 1, 5], then # `remaining` is [9, 6, 5]. remaining = np.cumsum(colors[::-1])[::-1] for i in range(len(colors)-1): if nsample < 1: break variate[i] = hypergeometric(colors[i], remaining[i+1], nsample) nsample -= variate[i] variate[-1] = nsample The default method is "marginals". For some cases (e.g. when `colors` contains relatively small integers), the "count" method can be significantly faster than the "marginals" method. If performance of the algorithm is important, test the two methods with typical inputs to decide which works best. .. versionadded:: 1.18.0 Examples -------- >>> colors = [16, 8, 4] >>> seed = 4861946401452 >>> gen = np.random.Generator(np.random.PCG64(seed)) >>> gen.multivariate_hypergeometric(colors, 6) array([5, 0, 1]) >>> gen.multivariate_hypergeometric(colors, 6, size=3) array([[5, 0, 1], [2, 2, 2], [3, 3, 0]]) >>> gen.multivariate_hypergeometric(colors, 6, size=(2, 2)) array([[[3, 2, 1], [3, 2, 1]], [[4, 1, 1], [3, 2, 1]]]) multinomial(n, pvals, size=None) Draw samples from a multinomial distribution. The multinomial distribution is a multivariate generalization of the binomial distribution. Take an experiment with one of ``p`` possible outcomes. An example of such an experiment is throwing a dice, where the outcome can be 1 through 6. Each sample drawn from the distribution represents `n` such experiments. Its values, ``X_i = [X_0, X_1, ..., X_p]``, represent the number of times the outcome was ``i``. Parameters ---------- n : int or array-like of ints Number of experiments. pvals : array-like of floats Probabilities of each of the ``p`` different outcomes with shape ``(k0, k1, ..., kn, p)``. Each element ``pvals[i,j,...,:]`` must sum to 1 (however, the last element is always assumed to account for the remaining probability, as long as ``sum(pvals[..., :-1], axis=-1) <= 1.0``. Must have at least 1 dimension where pvals.shape[-1] > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn each with ``p`` elements. Default is None where the output size is determined by the broadcast shape of ``n`` and all by the final dimension of ``pvals``, which is denoted as ``b=(b0, b1, ..., bq)``. If size is not None, then it must be compatible with the broadcast shape ``b``. Specifically, size must have ``q`` or more elements and size[-(q-j):] must equal ``bj``. Returns ------- out : ndarray The drawn samples, of shape size, if provided. When size is provided, the output shape is size + (p,) If not specified, the shape is determined by the broadcast shape of ``n`` and ``pvals``, ``(b0, b1, ..., bq)`` augmented with the dimension of the multinomial, ``p``, so that that output shape is ``(b0, b1, ..., bq, p)``. Each entry ``out[i,j,...,:]`` is a ``p``-dimensional value drawn from the distribution. .. versionchanged:: 1.22.0 Added support for broadcasting `pvals` against `n` Examples -------- Throw a dice 20 times: >>> rng = np.random.default_rng() >>> rng.multinomial(20, [1/6.]*6, size=1) array([[4, 1, 7, 5, 2, 1]]) # random It landed 4 times on 1, once on 2, etc. Now, throw the dice 20 times, and 20 times again: >>> rng.multinomial(20, [1/6.]*6, size=2) array([[3, 4, 3, 3, 4, 3], [2, 4, 3, 4, 0, 7]]) # random For the first run, we threw 3 times 1, 4 times 2, etc. For the second, we threw 2 times 1, 4 times 2, etc. Now, do one experiment throwing the dice 10 time, and 10 times again, and another throwing the dice 20 times, and 20 times again: >>> rng.multinomial([[10], [20]], [1/6.]*6, size=(2, 2)) array([[[2, 4, 0, 1, 2, 1], [1, 3, 0, 3, 1, 2]], [[1, 4, 4, 4, 4, 3], [3, 3, 2, 5, 5, 2]]]) # random The first array shows the outcomes of throwing the dice 10 times, and the second shows the outcomes from throwing the dice 20 times. A loaded die is more likely to land on number 6: >>> rng.multinomial(100, [1/7.]*5 + [2/7.]) array([11, 16, 14, 17, 16, 26]) # random Simulate 10 throws of a 4-sided die and 20 throws of a 6-sided die >>> rng.multinomial([10, 20],[[1/4]*4 + [0]*2, [1/6]*6]) array([[2, 1, 4, 3, 0, 0], [3, 3, 3, 6, 1, 4]], dtype=int64) # random Generate categorical random variates from two categories where the first has 3 outcomes and the second has 2. >>> rng.multinomial(1, [[.1, .5, .4 ], [.3, .7, .0]]) array([[0, 0, 1], [0, 1, 0]], dtype=int64) # random ``argmax(axis=-1)`` is then used to return the categories. >>> pvals = [[.1, .5, .4 ], [.3, .7, .0]] >>> rvs = rng.multinomial(1, pvals, size=(4,2)) >>> rvs.argmax(axis=-1) array([[0, 1], [2, 0], [2, 1], [2, 0]], dtype=int64) # random The same output dimension can be produced using broadcasting. >>> rvs = rng.multinomial([[1]] * 4, pvals) >>> rvs.argmax(axis=-1) array([[0, 1], [2, 0], [2, 1], [2, 0]], dtype=int64) # random The probability inputs should be normalized. As an implementation detail, the value of the last entry is ignored and assumed to take up any leftover probability mass, but this should not be relied on. A biased coin which has twice as much weight on one side as on the other should be sampled like so: >>> rng.multinomial(100, [1.0 / 3, 2.0 / 3]) # RIGHT array([38, 62]) # random not like: >>> rng.multinomial(100, [1.0, 2.0]) # WRONG Traceback (most recent call last): ValueError: pvals < 0, pvals > 1 or pvals contains NaNs multivariate_normal(mean, cov, size=None, check_valid='warn', tol=1e-8, *, method='svd') Draw random samples from a multivariate normal distribution. The multivariate normal, multinormal or Gaussian distribution is a generalization of the one-dimensional normal distribution to higher dimensions. Such a distribution is specified by its mean and covariance matrix. These parameters are analogous to the mean (average or "center") and variance (the squared standard deviation, or "width") of the one-dimensional normal distribution. Parameters ---------- mean : 1-D array_like, of length N Mean of the N-dimensional distribution. cov : 2-D array_like, of shape (N, N) Covariance matrix of the distribution. It must be symmetric and positive-semidefinite for proper sampling. size : int or tuple of ints, optional Given a shape of, for example, ``(m,n,k)``, ``m*n*k`` samples are generated, and packed in an `m`-by-`n`-by-`k` arrangement. Because each sample is `N`-dimensional, the output shape is ``(m,n,k,N)``. If no shape is specified, a single (`N`-D) sample is returned. check_valid : { 'warn', 'raise', 'ignore' }, optional Behavior when the covariance matrix is not positive semidefinite. tol : float, optional Tolerance when checking the singular values in covariance matrix. cov is cast to double before the check. method : { 'svd', 'eigh', 'cholesky'}, optional The cov input is used to compute a factor matrix A such that ``A @ A.T = cov``. This argument is used to select the method used to compute the factor matrix A. The default method 'svd' is the slowest, while 'cholesky' is the fastest but less robust than the slowest method. The method `eigh` uses eigen decomposition to compute A and is faster than svd but slower than cholesky. .. versionadded:: 1.18.0 Returns ------- out : ndarray The drawn samples, of shape *size*, if that was provided. If not, the shape is ``(N,)``. In other words, each entry ``out[i,j,...,:]`` is an N-dimensional value drawn from the distribution. Notes ----- The mean is a coordinate in N-dimensional space, which represents the location where samples are most likely to be generated. This is analogous to the peak of the bell curve for the one-dimensional or univariate normal distribution. Covariance indicates the level to which two variables vary together. From the multivariate normal distribution, we draw N-dimensional samples, :math:`X = [x_1, x_2, ... x_N]`. The covariance matrix element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`. The element :math:`C_{ii}` is the variance of :math:`x_i` (i.e. its "spread"). Instead of specifying the full covariance matrix, popular approximations include: - Spherical covariance (`cov` is a multiple of the identity matrix) - Diagonal covariance (`cov` has non-negative elements, and only on the diagonal) This geometrical property can be seen in two dimensions by plotting generated data-points: >>> mean = [0, 0] >>> cov = [[1, 0], [0, 100]] # diagonal covariance Diagonal covariance means that points are oriented along x or y-axis: >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() >>> x, y = rng.multivariate_normal(mean, cov, 5000).T >>> plt.plot(x, y, 'x') >>> plt.axis('equal') >>> plt.show() Note that the covariance matrix must be positive semidefinite (a.k.a. nonnegative-definite). Otherwise, the behavior of this method is undefined and backwards compatibility is not guaranteed. This function internally uses linear algebra routines, and thus results may not be identical (even up to precision) across architectures, OSes, or even builds. For example, this is likely if ``cov`` has multiple equal singular values and ``method`` is ``'svd'`` (default). In this case, ``method='cholesky'`` may be more robust. References ---------- .. [1] Papoulis, A., "Probability, Random Variables, and Stochastic Processes," 3rd ed., New York: McGraw-Hill, 1991. .. [2] Duda, R. O., Hart, P. E., and Stork, D. G., "Pattern Classification," 2nd ed., New York: Wiley, 2001. Examples -------- >>> mean = (1, 2) >>> cov = [[1, 0], [0, 1]] >>> rng = np.random.default_rng() >>> x = rng.multivariate_normal(mean, cov, (3, 3)) >>> x.shape (3, 3, 2) We can use a different method other than the default to factorize cov: >>> y = rng.multivariate_normal(mean, cov, (3, 3), method='cholesky') >>> y.shape (3, 3, 2) Here we generate 800 samples from the bivariate normal distribution with mean [0, 0] and covariance matrix [[6, -3], [-3, 3.5]]. The expected variances of the first and second components of the sample are 6 and 3.5, respectively, and the expected correlation coefficient is -3/sqrt(6*3.5) ≈ -0.65465. >>> cov = np.array([[6, -3], [-3, 3.5]]) >>> pts = rng.multivariate_normal([0, 0], cov, size=800) Check that the mean, covariance, and correlation coefficient of the sample are close to the expected values: >>> pts.mean(axis=0) array([ 0.0326911 , -0.01280782]) # may vary >>> np.cov(pts.T) array([[ 5.96202397, -2.85602287], [-2.85602287, 3.47613949]]) # may vary >>> np.corrcoef(pts.T)[0, 1] -0.6273591314603949 # may vary We can visualize this data with a scatter plot. The orientation of the point cloud illustrates the negative correlation of the components of this sample. >>> import matplotlib.pyplot as plt >>> plt.plot(pts[:, 0], pts[:, 1], '.', alpha=0.5) >>> plt.axis('equal') >>> plt.grid() >>> plt.show() logseries(p, size=None) Draw samples from a logarithmic series distribution. Samples are drawn from a log series distribution with specified shape parameter, 0 <= ``p`` < 1. Parameters ---------- p : float or array_like of floats Shape parameter for the distribution. Must be in the range [0, 1). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``p`` is a scalar. Otherwise, ``np.array(p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized logarithmic series distribution. See Also -------- scipy.stats.logser : probability density function, distribution or cumulative density function, etc. Notes ----- The probability mass function for the Log Series distribution is .. math:: P(k) = \frac{-p^k}{k \ln(1-p)}, where p = probability. The log series distribution is frequently used to represent species richness and occurrence, first proposed by Fisher, Corbet, and Williams in 1943 [2]. It may also be used to model the numbers of occupants seen in cars [3]. References ---------- .. [1] Buzas, Martin A.; Culver, Stephen J., Understanding regional species diversity through the log series distribution of occurrences: BIODIVERSITY RESEARCH Diversity & Distributions, Volume 5, Number 5, September 1999 , pp. 187-195(9). .. [2] Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology, 12:42-58. .. [3] D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small Data Sets, CRC Press, 1994. .. [4] Wikipedia, "Logarithmic distribution", https://en.wikipedia.org/wiki/Logarithmic_distribution Examples -------- Draw samples from the distribution: >>> a = .6 >>> rng = np.random.default_rng() >>> s = rng.logseries(a, 10000) >>> import matplotlib.pyplot as plt >>> bins = np.arange(-.5, max(s) + .5 ) >>> count, bins, _ = plt.hist(s, bins=bins, label='Sample count') # plot against distribution >>> def logseries(k, p): ... return -p**k/(k*np.log(1-p)) >>> centres = np.arange(1, max(s) + 1) >>> plt.plot(centres, logseries(centres, a) * s.size, 'r', label='logseries PMF') >>> plt.legend() >>> plt.show() hypergeometric(ngood, nbad, nsample, size=None) Draw samples from a Hypergeometric distribution. Samples are drawn from a hypergeometric distribution with specified parameters, `ngood` (ways to make a good selection), `nbad` (ways to make a bad selection), and `nsample` (number of items sampled, which is less than or equal to the sum ``ngood + nbad``). Parameters ---------- ngood : int or array_like of ints Number of ways to make a good selection. Must be nonnegative and less than 10**9. nbad : int or array_like of ints Number of ways to make a bad selection. Must be nonnegative and less than 10**9. nsample : int or array_like of ints Number of items sampled. Must be nonnegative and less than ``ngood + nbad``. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if `ngood`, `nbad`, and `nsample` are all scalars. Otherwise, ``np.broadcast(ngood, nbad, nsample).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized hypergeometric distribution. Each sample is the number of good items within a randomly selected subset of size `nsample` taken from a set of `ngood` good items and `nbad` bad items. See Also -------- multivariate_hypergeometric : Draw samples from the multivariate hypergeometric distribution. scipy.stats.hypergeom : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Hypergeometric distribution is .. math:: P(x) = \frac{\binom{g}{x}\binom{b}{n-x}}{\binom{g+b}{n}}, where :math:`0 \le x \le n` and :math:`n-b \le x \le g` for P(x) the probability of ``x`` good results in the drawn sample, g = `ngood`, b = `nbad`, and n = `nsample`. Consider an urn with black and white marbles in it, `ngood` of them are black and `nbad` are white. If you draw `nsample` balls without replacement, then the hypergeometric distribution describes the distribution of black balls in the drawn sample. Note that this distribution is very similar to the binomial distribution, except that in this case, samples are drawn without replacement, whereas in the Binomial case samples are drawn with replacement (or the sample space is infinite). As the sample space becomes large, this distribution approaches the binomial. The arguments `ngood` and `nbad` each must be less than `10**9`. For extremely large arguments, the algorithm that is used to compute the samples [4]_ breaks down because of loss of precision in floating point calculations. For such large values, if `nsample` is not also large, the distribution can be approximated with the binomial distribution, `binomial(n=nsample, p=ngood/(ngood + nbad))`. References ---------- .. [1] Lentner, Marvin, "Elementary Applied Statistics", Bogden and Quigley, 1972. .. [2] Weisstein, Eric W. "Hypergeometric Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HypergeometricDistribution.html .. [3] Wikipedia, "Hypergeometric distribution", https://en.wikipedia.org/wiki/Hypergeometric_distribution .. [4] Stadlober, Ernst, "The ratio of uniforms approach for generating discrete random variates", Journal of Computational and Applied Mathematics, 31, pp. 181-189 (1990). Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> ngood, nbad, nsamp = 100, 2, 10 # number of good, number of bad, and number of samples >>> s = rng.hypergeometric(ngood, nbad, nsamp, 1000) >>> from matplotlib.pyplot import hist >>> hist(s) # note that it is very unlikely to grab both bad items Suppose you have an urn with 15 white and 15 black marbles. If you pull 15 marbles at random, how likely is it that 12 or more of them are one color? >>> s = rng.hypergeometric(15, 15, 15, 100000) >>> sum(s>=12)/100000. + sum(s<=3)/100000. # answer = 0.003 ... pretty unlikely! geometric(p, size=None) Draw samples from the geometric distribution. Bernoulli trials are experiments with one of two outcomes: success or failure (an example of such an experiment is flipping a coin). The geometric distribution models the number of trials that must be run in order to achieve success. It is therefore supported on the positive integers, ``k = 1, 2, ...``. The probability mass function of the geometric distribution is .. math:: f(k) = (1 - p)^{k - 1} p where `p` is the probability of success of an individual trial. Parameters ---------- p : float or array_like of floats The probability of success of an individual trial. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``p`` is a scalar. Otherwise, ``np.array(p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized geometric distribution. References ---------- .. [1] Wikipedia, "Geometric distribution", https://en.wikipedia.org/wiki/Geometric_distribution Examples -------- Draw 10,000 values from the geometric distribution, with the probability of an individual success equal to ``p = 0.35``: >>> p, size = 0.35, 10000 >>> rng = np.random.default_rng() >>> sample = rng.geometric(p=p, size=size) What proportion of trials succeeded after a single run? >>> (sample == 1).sum()/size 0.34889999999999999 # may vary The geometric distribution with ``p=0.35`` looks as follows: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(sample, bins=30, density=True) >>> plt.plot(bins, (1-p)**(bins-1)*p) >>> plt.xlim([0, 25]) >>> plt.show() zipf(a, size=None) Draw samples from a Zipf distribution. Samples are drawn from a Zipf distribution with specified parameter `a` > 1. The Zipf distribution (also known as the zeta distribution) is a discrete probability distribution that satisfies Zipf's law: the frequency of an item is inversely proportional to its rank in a frequency table. Parameters ---------- a : float or array_like of floats Distribution parameter. Must be greater than 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Zipf distribution. See Also -------- scipy.stats.zipf : probability density function, distribution, or cumulative density function, etc. Notes ----- The probability density for the Zipf distribution is .. math:: p(k) = \frac{k^{-a}}{\zeta(a)}, for integers :math:`k \geq 1`, where :math:`\zeta` is the Riemann Zeta function. It is named for the American linguist George Kingsley Zipf, who noted that the frequency of any word in a sample of a language is inversely proportional to its rank in the frequency table. References ---------- .. [1] Zipf, G. K., "Selected Studies of the Principle of Relative Frequency in Language," Cambridge, MA: Harvard Univ. Press, 1932. Examples -------- Draw samples from the distribution: >>> a = 4.0 >>> n = 20000 >>> rng = np.random.default_rng() >>> s = rng.zipf(a, size=n) Display the histogram of the samples, along with the expected histogram based on the probability density function: >>> import matplotlib.pyplot as plt >>> from scipy.special import zeta # doctest: +SKIP `bincount` provides a fast histogram for small integers. >>> count = np.bincount(s) >>> k = np.arange(1, s.max() + 1) >>> plt.bar(k, count[1:], alpha=0.5, label='sample count') >>> plt.plot(k, n*(k**-a)/zeta(a), 'k.-', alpha=0.5, ... label='expected count') # doctest: +SKIP >>> plt.semilogy() >>> plt.grid(alpha=0.4) >>> plt.legend() >>> plt.title(f'Zipf sample, a={a}, size={n}') >>> plt.show() poisson(lam=1.0, size=None) Draw samples from a Poisson distribution. The Poisson distribution is the limit of the binomial distribution for large N. Parameters ---------- lam : float or array_like of floats Expected number of events occurring in a fixed-time interval, must be >= 0. A sequence must be broadcastable over the requested size. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``lam`` is a scalar. Otherwise, ``np.array(lam).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Poisson distribution. Notes ----- The Poisson distribution .. math:: f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!} For events with an expected separation :math:`\lambda` the Poisson distribution :math:`f(k; \lambda)` describes the probability of :math:`k` events occurring within the observed interval :math:`\lambda`. Because the output is limited to the range of the C int64 type, a ValueError is raised when `lam` is within 10 sigma of the maximum representable value. References ---------- .. [1] Weisstein, Eric W. "Poisson Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PoissonDistribution.html .. [2] Wikipedia, "Poisson distribution", https://en.wikipedia.org/wiki/Poisson_distribution Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> lam, size = 5, 10000 >>> s = rng.poisson(lam=lam, size=size) Verify the mean and variance, which should be approximately ``lam``: >>> s.mean(), s.var() (4.9917 5.1088311) # may vary Display the histogram and probability mass function: >>> import matplotlib.pyplot as plt >>> from scipy import stats >>> x = np.arange(0, 21) >>> pmf = stats.poisson.pmf(x, mu=lam) >>> plt.hist(s, bins=x, density=True, width=0.5) >>> plt.stem(x, pmf, 'C1-') >>> plt.show() Draw each 100 values for lambda 100 and 500: >>> s = rng.poisson(lam=(100., 500.), size=(100, 2)) negative_binomial(n, p, size=None) Draw samples from a negative binomial distribution. Samples are drawn from a negative binomial distribution with specified parameters, `n` successes and `p` probability of success where `n` is > 0 and `p` is in the interval (0, 1]. Parameters ---------- n : float or array_like of floats Parameter of the distribution, > 0. p : float or array_like of floats Parameter of the distribution. Must satisfy 0 < p <= 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``n`` and ``p`` are both scalars. Otherwise, ``np.broadcast(n, p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized negative binomial distribution, where each sample is equal to N, the number of failures that occurred before a total of n successes was reached. Notes ----- The probability mass function of the negative binomial distribution is .. math:: P(N;n,p) = \frac{\Gamma(N+n)}{N!\Gamma(n)}p^{n}(1-p)^{N}, where :math:`n` is the number of successes, :math:`p` is the probability of success, :math:`N+n` is the number of trials, and :math:`\Gamma` is the gamma function. When :math:`n` is an integer, :math:`\frac{\Gamma(N+n)}{N!\Gamma(n)} = \binom{N+n-1}{N}`, which is the more common form of this term in the pmf. The negative binomial distribution gives the probability of N failures given n successes, with a success on the last trial. If one throws a die repeatedly until the third time a "1" appears, then the probability distribution of the number of non-"1"s that appear before the third "1" is a negative binomial distribution. Because this method internally calls ``Generator.poisson`` with an intermediate random value, a ValueError is raised when the choice of :math:`n` and :math:`p` would result in the mean + 10 sigma of the sampled intermediate distribution exceeding the max acceptable value of the ``Generator.poisson`` method. This happens when :math:`p` is too low (a lot of failures happen for every success) and :math:`n` is too big ( a lot of successes are allowed). Therefore, the :math:`n` and :math:`p` values must satisfy the constraint: .. math:: n\frac{1-p}{p}+10n\sqrt{n}\frac{1-p}{p}<2^{63}-1-10\sqrt{2^{63}-1}, Where the left side of the equation is the derived mean + 10 sigma of a sample from the gamma distribution internally used as the :math:`lam` parameter of a poisson sample, and the right side of the equation is the constraint for maximum value of :math:`lam` in ``Generator.poisson``. References ---------- .. [1] Weisstein, Eric W. "Negative Binomial Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NegativeBinomialDistribution.html .. [2] Wikipedia, "Negative binomial distribution", https://en.wikipedia.org/wiki/Negative_binomial_distribution Examples -------- Draw samples from the distribution: A real world example. A company drills wild-cat oil exploration wells, each with an estimated probability of success of 0.1. What is the probability of having one success for each successive well, that is what is the probability of a single success after drilling 5 wells, after 6 wells, etc.? >>> rng = np.random.default_rng() >>> s = rng.negative_binomial(1, 0.1, 100000) >>> for i in range(1, 11): # doctest: +SKIP ... probability = sum(s>> rng = np.random.default_rng() >>> n, p, size = 10, .5, 10000 >>> s = rng.binomial(n, p, 10000) Assume a company drills 9 wild-cat oil exploration wells, each with an estimated probability of success of ``p=0.1``. All nine wells fail. What is the probability of that happening? Over ``size = 20,000`` trials the probability of this happening is on average: >>> n, p, size = 9, 0.1, 20000 >>> np.sum(rng.binomial(n=n, p=p, size=size) == 0)/size 0.39015 # may vary The following can be used to visualize a sample with ``n=100``, ``p=0.4`` and the corresponding probability density function: >>> import matplotlib.pyplot as plt >>> from scipy.stats import binom >>> n, p, size = 100, 0.4, 10000 >>> sample = rng.binomial(n, p, size=size) >>> count, bins, _ = plt.hist(sample, 30, density=True) >>> x = np.arange(n) >>> y = binom.pmf(x, n, p) >>> plt.plot(x, y, linewidth=2, color='r') triangular(left, mode, right, size=None) Draw samples from the triangular distribution over the interval ``[left, right]``. The triangular distribution is a continuous probability distribution with lower limit left, peak at mode, and upper limit right. Unlike the other distributions, these parameters directly define the shape of the pdf. Parameters ---------- left : float or array_like of floats Lower limit. mode : float or array_like of floats The value where the peak of the distribution occurs. The value must fulfill the condition ``left <= mode <= right``. right : float or array_like of floats Upper limit, must be larger than `left`. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``left``, ``mode``, and ``right`` are all scalars. Otherwise, ``np.broadcast(left, mode, right).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized triangular distribution. Notes ----- The probability density function for the triangular distribution is .. math:: P(x;l, m, r) = \begin{cases} \frac{2(x-l)}{(r-l)(m-l)}& \text{for $l \leq x \leq m$},\\ \frac{2(r-x)}{(r-l)(r-m)}& \text{for $m \leq x \leq r$},\\ 0& \text{otherwise}. \end{cases} The triangular distribution is often used in ill-defined problems where the underlying distribution is not known, but some knowledge of the limits and mode exists. Often it is used in simulations. References ---------- .. [1] Wikipedia, "Triangular distribution" https://en.wikipedia.org/wiki/Triangular_distribution Examples -------- Draw values from the distribution and plot the histogram: >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() >>> h = plt.hist(rng.triangular(-3, 0, 8, 100000), bins=200, ... density=True) >>> plt.show() wald(mean, scale, size=None) Draw samples from a Wald, or inverse Gaussian, distribution. As the scale approaches infinity, the distribution becomes more like a Gaussian. Some references claim that the Wald is an inverse Gaussian with mean equal to 1, but this is by no means universal. The inverse Gaussian distribution was first studied in relationship to Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian because there is an inverse relationship between the time to cover a unit distance and distance covered in unit time. Parameters ---------- mean : float or array_like of floats Distribution mean, must be > 0. scale : float or array_like of floats Scale parameter, must be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``mean`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(mean, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Wald distribution. Notes ----- The probability density function for the Wald distribution is .. math:: P(x;mean,scale) = \sqrt{\frac{scale}{2\pi x^3}}e^ \frac{-scale(x-mean)^2}{2\cdotp mean^2x} As noted above the inverse Gaussian distribution first arise from attempts to model Brownian motion. It is also a competitor to the Weibull for use in reliability modeling and modeling stock returns and interest rate processes. References ---------- .. [1] Brighton Webs Ltd., Wald Distribution, https://web.archive.org/web/20090423014010/http://www.brighton-webs.co.uk:80/distributions/wald.asp .. [2] Chhikara, Raj S., and Folks, J. Leroy, "The Inverse Gaussian Distribution: Theory : Methodology, and Applications", CRC Press, 1988. .. [3] Wikipedia, "Inverse Gaussian distribution" https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Examples -------- Draw values from the distribution and plot the histogram: >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() >>> h = plt.hist(rng.wald(3, 2, 100000), bins=200, density=True) >>> plt.show() rayleigh(scale=1.0, size=None) Draw samples from a Rayleigh distribution. The :math:`\chi` and Weibull distributions are generalizations of the Rayleigh. Parameters ---------- scale : float or array_like of floats, optional Scale, also equals the mode. Must be non-negative. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``scale`` is a scalar. Otherwise, ``np.array(scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Rayleigh distribution. Notes ----- The probability density function for the Rayleigh distribution is .. math:: P(x;scale) = \frac{x}{scale^2}e^{\frac{-x^2}{2 \cdotp scale^2}} The Rayleigh distribution would arise, for example, if the East and North components of the wind velocity had identical zero-mean Gaussian distributions. Then the wind speed would have a Rayleigh distribution. References ---------- .. [1] Brighton Webs Ltd., "Rayleigh Distribution," https://web.archive.org/web/20090514091424/http://brighton-webs.co.uk:80/distributions/rayleigh.asp .. [2] Wikipedia, "Rayleigh distribution" https://en.wikipedia.org/wiki/Rayleigh_distribution Examples -------- Draw values from the distribution and plot the histogram >>> from matplotlib.pyplot import hist >>> rng = np.random.default_rng() >>> values = hist(rng.rayleigh(3, 100000), bins=200, density=True) Wave heights tend to follow a Rayleigh distribution. If the mean wave height is 1 meter, what fraction of waves are likely to be larger than 3 meters? >>> meanvalue = 1 >>> modevalue = np.sqrt(2 / np.pi) * meanvalue >>> s = rng.rayleigh(modevalue, 1000000) The percentage of waves larger than 3 meters is: >>> 100.*sum(s>3)/1000000. 0.087300000000000003 # random lognormal(mean=0.0, sigma=1.0, size=None) Draw samples from a log-normal distribution. Draw samples from a log-normal distribution with specified mean, standard deviation, and array shape. Note that the mean and standard deviation are not the values for the distribution itself, but of the underlying normal distribution it is derived from. Parameters ---------- mean : float or array_like of floats, optional Mean value of the underlying normal distribution. Default is 0. sigma : float or array_like of floats, optional Standard deviation of the underlying normal distribution. Must be non-negative. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``mean`` and ``sigma`` are both scalars. Otherwise, ``np.broadcast(mean, sigma).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized log-normal distribution. See Also -------- scipy.stats.lognorm : probability density function, distribution, cumulative density function, etc. Notes ----- A variable `x` has a log-normal distribution if `log(x)` is normally distributed. The probability density function for the log-normal distribution is: .. math:: p(x) = \frac{1}{\sigma x \sqrt{2\pi}} e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})} where :math:`\mu` is the mean and :math:`\sigma` is the standard deviation of the normally distributed logarithm of the variable. A log-normal distribution results if a random variable is the *product* of a large number of independent, identically-distributed variables in the same way that a normal distribution results if the variable is the *sum* of a large number of independent, identically-distributed variables. References ---------- .. [1] Limpert, E., Stahel, W. A., and Abbt, M., "Log-normal Distributions across the Sciences: Keys and Clues," BioScience, Vol. 51, No. 5, May, 2001. https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf .. [2] Reiss, R.D. and Thomas, M., "Statistical Analysis of Extreme Values," Basel: Birkhauser Verlag, 2001, pp. 31-32. Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> mu, sigma = 3., 1. # mean and standard deviation >>> s = rng.lognormal(mu, sigma, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, 100, density=True, align='mid') >>> x = np.linspace(min(bins), max(bins), 10000) >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2)) ... / (x * sigma * np.sqrt(2 * np.pi))) >>> plt.plot(x, pdf, linewidth=2, color='r') >>> plt.axis('tight') >>> plt.show() Demonstrate that taking the products of random samples from a uniform distribution can be fit well by a log-normal probability density function. >>> # Generate a thousand samples: each is the product of 100 random >>> # values, drawn from a normal distribution. >>> rng = rng >>> b = [] >>> for i in range(1000): ... a = 10. + rng.standard_normal(100) ... b.append(np.prod(a)) >>> b = np.array(b) / np.min(b) # scale values to be positive >>> count, bins, _ = plt.hist(b, 100, density=True, align='mid') >>> sigma = np.std(np.log(b)) >>> mu = np.mean(np.log(b)) >>> x = np.linspace(min(bins), max(bins), 10000) >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2)) ... / (x * sigma * np.sqrt(2 * np.pi))) >>> plt.plot(x, pdf, color='r', linewidth=2) >>> plt.show() logistic(loc=0.0, scale=1.0, size=None) Draw samples from a logistic distribution. Samples are drawn from a logistic distribution with specified parameters, loc (location or mean, also median), and scale (>0). Parameters ---------- loc : float or array_like of floats, optional Parameter of the distribution. Default is 0. scale : float or array_like of floats, optional Parameter of the distribution. Must be non-negative. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized logistic distribution. See Also -------- scipy.stats.logistic : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Logistic distribution is .. math:: P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2}, where :math:`\mu` = location and :math:`s` = scale. The Logistic distribution is used in Extreme Value problems where it can act as a mixture of Gumbel distributions, in Epidemiology, and by the World Chess Federation (FIDE) where it is used in the Elo ranking system, assuming the performance of each player is a logistically distributed random variable. References ---------- .. [1] Reiss, R.-D. and Thomas M. (2001), "Statistical Analysis of Extreme Values, from Insurance, Finance, Hydrology and Other Fields," Birkhauser Verlag, Basel, pp 132-133. .. [2] Weisstein, Eric W. "Logistic Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LogisticDistribution.html .. [3] Wikipedia, "Logistic-distribution", https://en.wikipedia.org/wiki/Logistic_distribution Examples -------- Draw samples from the distribution: >>> loc, scale = 10, 1 >>> rng = np.random.default_rng() >>> s = rng.logistic(loc, scale, 10000) >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, bins=50, label='Sampled data') # plot sampled data against the exact distribution >>> def logistic(x, loc, scale): ... return np.exp((loc-x)/scale)/(scale*(1+np.exp((loc-x)/scale))**2) >>> logistic_values = logistic(bins, loc, scale) >>> bin_spacing = np.mean(np.diff(bins)) >>> plt.plot(bins, logistic_values * bin_spacing * s.size, label='Logistic PDF') >>> plt.legend() >>> plt.show() gumbel(loc=0.0, scale=1.0, size=None) Draw samples from a Gumbel distribution. Draw samples from a Gumbel distribution with specified location and scale. For more information on the Gumbel distribution, see Notes and References below. Parameters ---------- loc : float or array_like of floats, optional The location of the mode of the distribution. Default is 0. scale : float or array_like of floats, optional The scale parameter of the distribution. Default is 1. Must be non- negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Gumbel distribution. See Also -------- scipy.stats.gumbel_l scipy.stats.gumbel_r scipy.stats.genextreme weibull Notes ----- The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme Value Type I) distribution is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. The Gumbel is a special case of the Extreme Value Type I distribution for maximums from distributions with "exponential-like" tails. The probability density for the Gumbel distribution is .. math:: p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/ \beta}}, where :math:`\mu` is the mode, a location parameter, and :math:`\beta` is the scale parameter. The Gumbel (named for German mathematician Emil Julius Gumbel) was used very early in the hydrology literature, for modeling the occurrence of flood events. It is also used for modeling maximum wind speed and rainfall rates. It is a "fat-tailed" distribution - the probability of an event in the tail of the distribution is larger than if one used a Gaussian, hence the surprisingly frequent occurrence of 100-year floods. Floods were initially modeled as a Gaussian process, which underestimated the frequency of extreme events. It is one of a class of extreme value distributions, the Generalized Extreme Value (GEV) distributions, which also includes the Weibull and Frechet. The function has a mean of :math:`\mu + 0.57721\beta` and a variance of :math:`\frac{\pi^2}{6}\beta^2`. References ---------- .. [1] Gumbel, E. J., "Statistics of Extremes," New York: Columbia University Press, 1958. .. [2] Reiss, R.-D. and Thomas, M., "Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields," Basel: Birkhauser Verlag, 2001. Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> mu, beta = 0, 0.1 # location and scale >>> s = rng.gumbel(mu, beta, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, 30, density=True) >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta) ... * np.exp( -np.exp( -(bins - mu) /beta) ), ... linewidth=2, color='r') >>> plt.show() Show how an extreme value distribution can arise from a Gaussian process and compare to a Gaussian: >>> means = [] >>> maxima = [] >>> for i in range(0,1000) : ... a = rng.normal(mu, beta, 1000) ... means.append(a.mean()) ... maxima.append(a.max()) >>> count, bins, _ = plt.hist(maxima, 30, density=True) >>> beta = np.std(maxima) * np.sqrt(6) / np.pi >>> mu = np.mean(maxima) - 0.57721*beta >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta) ... * np.exp(-np.exp(-(bins - mu)/beta)), ... linewidth=2, color='r') >>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi)) ... * np.exp(-(bins - mu)**2 / (2 * beta**2)), ... linewidth=2, color='g') >>> plt.show() laplace(loc=0.0, scale=1.0, size=None) Draw samples from the Laplace or double exponential distribution with specified location (or mean) and scale (decay). The Laplace distribution is similar to the Gaussian/normal distribution, but is sharper at the peak and has fatter tails. It represents the difference between two independent, identically distributed exponential random variables. Parameters ---------- loc : float or array_like of floats, optional The position, :math:`\mu`, of the distribution peak. Default is 0. scale : float or array_like of floats, optional :math:`\lambda`, the exponential decay. Default is 1. Must be non- negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Laplace distribution. Notes ----- It has the probability density function .. math:: f(x; \mu, \lambda) = \frac{1}{2\lambda} \exp\left(-\frac{|x - \mu|}{\lambda}\right). The first law of Laplace, from 1774, states that the frequency of an error can be expressed as an exponential function of the absolute magnitude of the error, which leads to the Laplace distribution. For many problems in economics and health sciences, this distribution seems to model the data better than the standard Gaussian distribution. References ---------- .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing," New York: Dover, 1972. .. [2] Kotz, Samuel, et. al. "The Laplace Distribution and Generalizations, " Birkhauser, 2001. .. [3] Weisstein, Eric W. "Laplace Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LaplaceDistribution.html .. [4] Wikipedia, "Laplace distribution", https://en.wikipedia.org/wiki/Laplace_distribution Examples -------- Draw samples from the distribution >>> loc, scale = 0., 1. >>> rng = np.random.default_rng() >>> s = rng.laplace(loc, scale, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, 30, density=True) >>> x = np.arange(-8., 8., .01) >>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale) >>> plt.plot(x, pdf) Plot Gaussian for comparison: >>> g = (1/(scale * np.sqrt(2 * np.pi)) * ... np.exp(-(x - loc)**2 / (2 * scale**2))) >>> plt.plot(x,g) power(a, size=None) Draws samples in [0, 1] from a power distribution with positive exponent a - 1. Also known as the power function distribution. Parameters ---------- a : float or array_like of floats Parameter of the distribution. Must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized power distribution. Raises ------ ValueError If a <= 0. Notes ----- The probability density function is .. math:: P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0. The power function distribution is just the inverse of the Pareto distribution. It may also be seen as a special case of the Beta distribution. It is used, for example, in modeling the over-reporting of insurance claims. References ---------- .. [1] Christian Kleiber, Samuel Kotz, "Statistical size distributions in economics and actuarial sciences", Wiley, 2003. .. [2] Heckert, N. A. and Filliben, James J. "NIST Handbook 148: Dataplot Reference Manual, Volume 2: Let Subcommands and Library Functions", National Institute of Standards and Technology Handbook Series, June 2003. https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> a = 5. # shape >>> samples = 1000 >>> s = rng.power(a, samples) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, bins=30) >>> x = np.linspace(0, 1, 100) >>> y = a*x**(a-1.) >>> normed_y = samples*np.diff(bins)[0]*y >>> plt.plot(x, normed_y) >>> plt.show() Compare the power function distribution to the inverse of the Pareto. >>> from scipy import stats # doctest: +SKIP >>> rvs = rng.power(5, 1000000) >>> rvsp = rng.pareto(5, 1000000) >>> xx = np.linspace(0,1,100) >>> powpdf = stats.powerlaw.pdf(xx,5) # doctest: +SKIP >>> plt.figure() >>> plt.hist(rvs, bins=50, density=True) >>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP >>> plt.title('power(5)') >>> plt.figure() >>> plt.hist(1./(1.+rvsp), bins=50, density=True) >>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP >>> plt.title('inverse of 1 + Generator.pareto(5)') >>> plt.figure() >>> plt.hist(1./(1.+rvsp), bins=50, density=True) >>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP >>> plt.title('inverse of stats.pareto(5)') weibull(a, size=None) Draw samples from a Weibull distribution. Draw samples from a 1-parameter Weibull distribution with the given shape parameter `a`. .. math:: X = (-ln(U))^{1/a} Here, U is drawn from the uniform distribution over (0,1]. The more common 2-parameter Weibull, including a scale parameter :math:`\lambda` is just :math:`X = \lambda(-ln(U))^{1/a}`. Parameters ---------- a : float or array_like of floats Shape parameter of the distribution. Must be nonnegative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Weibull distribution. See Also -------- scipy.stats.weibull_max scipy.stats.weibull_min scipy.stats.genextreme gumbel Notes ----- The Weibull (or Type III asymptotic extreme value distribution for smallest values, SEV Type III, or Rosin-Rammler distribution) is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. This class includes the Gumbel and Frechet distributions. The probability density for the Weibull distribution is .. math:: p(x) = \frac{a} {\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a}, where :math:`a` is the shape and :math:`\lambda` the scale. The function has its peak (the mode) at :math:`\lambda(\frac{a-1}{a})^{1/a}`. When ``a = 1``, the Weibull distribution reduces to the exponential distribution. References ---------- .. [1] Waloddi Weibull, Royal Technical University, Stockholm, 1939 "A Statistical Theory Of The Strength Of Materials", Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939, Generalstabens Litografiska Anstalts Forlag, Stockholm. .. [2] Waloddi Weibull, "A Statistical Distribution Function of Wide Applicability", Journal Of Applied Mechanics ASME Paper 1951. .. [3] Wikipedia, "Weibull distribution", https://en.wikipedia.org/wiki/Weibull_distribution Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> a = 5. # shape >>> s = rng.weibull(a, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> def weibull(x, n, a): ... return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a) >>> count, bins, _ = plt.hist(rng.weibull(5., 1000)) >>> x = np.linspace(0, 2, 1000) >>> bin_spacing = np.mean(np.diff(bins)) >>> plt.plot(x, weibull(x, 1., 5.) * bin_spacing * s.size, label='Weibull PDF') >>> plt.legend() >>> plt.show() pareto(a, size=None) Draw samples from a Pareto II (AKA Lomax) distribution with specified shape. Parameters ---------- a : float or array_like of floats Shape of the distribution. Must be positive. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the Pareto II distribution. See Also -------- scipy.stats.pareto : Pareto I distribution scipy.stats.lomax : Lomax (Pareto II) distribution scipy.stats.genpareto : Generalized Pareto distribution Notes ----- The probability density for the Pareto II distribution is .. math:: p(x) = \frac{a}{{x+1}^{a+1}} , x \ge 0 where :math:`a > 0` is the shape. The Pareto II distribution is a shifted and scaled version of the Pareto I distribution, which can be found in `scipy.stats.pareto`. References ---------- .. [1] Francis Hunt and Paul Johnson, On the Pareto Distribution of Sourceforge projects. .. [2] Pareto, V. (1896). Course of Political Economy. Lausanne. .. [3] Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme Values, Birkhauser Verlag, Basel, pp 23-30. .. [4] Wikipedia, "Pareto distribution", https://en.wikipedia.org/wiki/Pareto_distribution Examples -------- Draw samples from the distribution: >>> a = 3. >>> rng = np.random.default_rng() >>> s = rng.pareto(a, 10000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> x = np.linspace(0, 3, 50) >>> pdf = a / (x+1)**(a+1) >>> plt.hist(s, bins=x, density=True, label='histogram') >>> plt.plot(x, pdf, linewidth=2, color='r', label='pdf') >>> plt.xlim(x.min(), x.max()) >>> plt.legend() >>> plt.show() vonmises(mu, kappa, size=None) Draw samples from a von Mises distribution. Samples are drawn from a von Mises distribution with specified mode (mu) and dispersion (kappa), on the interval [-pi, pi]. The von Mises distribution (also known as the circular normal distribution) is a continuous probability distribution on the unit circle. It may be thought of as the circular analogue of the normal distribution. Parameters ---------- mu : float or array_like of floats Mode ("center") of the distribution. kappa : float or array_like of floats Dispersion of the distribution, has to be >=0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``mu`` and ``kappa`` are both scalars. Otherwise, ``np.broadcast(mu, kappa).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized von Mises distribution. See Also -------- scipy.stats.vonmises : probability density function, distribution, or cumulative density function, etc. Notes ----- The probability density for the von Mises distribution is .. math:: p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}, where :math:`\mu` is the mode and :math:`\kappa` the dispersion, and :math:`I_0(\kappa)` is the modified Bessel function of order 0. The von Mises is named for Richard Edler von Mises, who was born in Austria-Hungary, in what is now the Ukraine. He fled to the United States in 1939 and became a professor at Harvard. He worked in probability theory, aerodynamics, fluid mechanics, and philosophy of science. References ---------- .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing," New York: Dover, 1972. .. [2] von Mises, R., "Mathematical Theory of Probability and Statistics", New York: Academic Press, 1964. Examples -------- Draw samples from the distribution: >>> mu, kappa = 0.0, 4.0 # mean and dispersion >>> rng = np.random.default_rng() >>> s = rng.vonmises(mu, kappa, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> from scipy.special import i0 # doctest: +SKIP >>> plt.hist(s, 50, density=True) >>> x = np.linspace(-np.pi, np.pi, num=51) >>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa)) # doctest: +SKIP >>> plt.plot(x, y, linewidth=2, color='r') # doctest: +SKIP >>> plt.show() standard_t(df, size=None) Draw samples from a standard Student's t distribution with `df` degrees of freedom. A special case of the hyperbolic distribution. As `df` gets large, the result resembles that of the standard normal distribution (`standard_normal`). Parameters ---------- df : float or array_like of floats Degrees of freedom, must be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``df`` is a scalar. Otherwise, ``np.array(df).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized standard Student's t distribution. Notes ----- The probability density function for the t distribution is .. math:: P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df} \Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2} The t test is based on an assumption that the data come from a Normal distribution. The t test provides a way to test whether the sample mean (that is the mean calculated from the data) is a good estimate of the true mean. The derivation of the t-distribution was first published in 1908 by William Gosset while working for the Guinness Brewery in Dublin. Due to proprietary issues, he had to publish under a pseudonym, and so he used the name Student. References ---------- .. [1] Dalgaard, Peter, "Introductory Statistics With R", Springer, 2002. .. [2] Wikipedia, "Student's t-distribution" https://en.wikipedia.org/wiki/Student's_t-distribution Examples -------- From Dalgaard page 83 [1]_, suppose the daily energy intake for 11 women in kilojoules (kJ) is: >>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \ ... 7515, 8230, 8770]) Does their energy intake deviate systematically from the recommended value of 7725 kJ? Our null hypothesis will be the absence of deviation, and the alternate hypothesis will be the presence of an effect that could be either positive or negative, hence making our test 2-tailed. Because we are estimating the mean and we have N=11 values in our sample, we have N-1=10 degrees of freedom. We set our significance level to 95% and compute the t statistic using the empirical mean and empirical standard deviation of our intake. We use a ddof of 1 to base the computation of our empirical standard deviation on an unbiased estimate of the variance (note: the final estimate is not unbiased due to the concave nature of the square root). >>> np.mean(intake) 6753.636363636364 >>> intake.std(ddof=1) 1142.1232221373727 >>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake))) >>> t -2.8207540608310198 We draw 1000000 samples from Student's t distribution with the adequate degrees of freedom. >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() >>> s = rng.standard_t(10, size=1000000) >>> h = plt.hist(s, bins=100, density=True) Does our t statistic land in one of the two critical regions found at both tails of the distribution? >>> np.sum(np.abs(t) < np.abs(s)) / float(len(s)) 0.018318 #random < 0.05, statistic is in critical region The probability value for this 2-tailed test is about 1.83%, which is lower than the 5% pre-determined significance threshold. Therefore, the probability of observing values as extreme as our intake conditionally on the null hypothesis being true is too low, and we reject the null hypothesis of no deviation. standard_cauchy(size=None) Draw samples from a standard Cauchy distribution with mode = 0. Also known as the Lorentz distribution. Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- samples : ndarray or scalar The drawn samples. Notes ----- The probability density function for the full Cauchy distribution is .. math:: P(x; x_0, \gamma) = \frac{1}{\pi \gamma \bigl[ 1+ (\frac{x-x_0}{\gamma})^2 \bigr] } and the Standard Cauchy distribution just sets :math:`x_0=0` and :math:`\gamma=1` The Cauchy distribution arises in the solution to the driven harmonic oscillator problem, and also describes spectral line broadening. It also describes the distribution of values at which a line tilted at a random angle will cut the x axis. When studying hypothesis tests that assume normality, seeing how the tests perform on data from a Cauchy distribution is a good indicator of their sensitivity to a heavy-tailed distribution, since the Cauchy looks very much like a Gaussian distribution, but with heavier tails. References ---------- .. [1] NIST/SEMATECH e-Handbook of Statistical Methods, "Cauchy Distribution", https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm .. [2] Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CauchyDistribution.html .. [3] Wikipedia, "Cauchy distribution" https://en.wikipedia.org/wiki/Cauchy_distribution Examples -------- Draw samples and plot the distribution: >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() >>> s = rng.standard_cauchy(1000000) >>> s = s[(s>-25) & (s<25)] # truncate distribution so it plots well >>> plt.hist(s, bins=100) >>> plt.show() noncentral_chisquare(df, nonc, size=None) Draw samples from a noncentral chi-square distribution. The noncentral :math:`\chi^2` distribution is a generalization of the :math:`\chi^2` distribution. Parameters ---------- df : float or array_like of floats Degrees of freedom, must be > 0. .. versionchanged:: 1.10.0 Earlier NumPy versions required dfnum > 1. nonc : float or array_like of floats Non-centrality, must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``df`` and ``nonc`` are both scalars. Otherwise, ``np.broadcast(df, nonc).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized noncentral chi-square distribution. Notes ----- The probability density function for the noncentral Chi-square distribution is .. math:: P(x;df,nonc) = \sum^{\infty}_{i=0} \frac{e^{-nonc/2}(nonc/2)^{i}}{i!} P_{Y_{df+2i}}(x), where :math:`Y_{q}` is the Chi-square with q degrees of freedom. References ---------- .. [1] Wikipedia, "Noncentral chi-squared distribution" https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution Examples -------- Draw values from the distribution and plot the histogram >>> rng = np.random.default_rng() >>> import matplotlib.pyplot as plt >>> values = plt.hist(rng.noncentral_chisquare(3, 20, 100000), ... bins=200, density=True) >>> plt.show() Draw values from a noncentral chisquare with very small noncentrality, and compare to a chisquare. >>> plt.figure() >>> values = plt.hist(rng.noncentral_chisquare(3, .0000001, 100000), ... bins=np.arange(0., 25, .1), density=True) >>> values2 = plt.hist(rng.chisquare(3, 100000), ... bins=np.arange(0., 25, .1), density=True) >>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob') >>> plt.show() Demonstrate how large values of non-centrality lead to a more symmetric distribution. >>> plt.figure() >>> values = plt.hist(rng.noncentral_chisquare(3, 20, 100000), ... bins=200, density=True) >>> plt.show() chisquare(df, size=None) Draw samples from a chi-square distribution. When `df` independent random variables, each with standard normal distributions (mean 0, variance 1), are squared and summed, the resulting distribution is chi-square (see Notes). This distribution is often used in hypothesis testing. Parameters ---------- df : float or array_like of floats Number of degrees of freedom, must be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``df`` is a scalar. Otherwise, ``np.array(df).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized chi-square distribution. Raises ------ ValueError When `df` <= 0 or when an inappropriate `size` (e.g. ``size=-1``) is given. Notes ----- The variable obtained by summing the squares of `df` independent, standard normally distributed random variables: .. math:: Q = \sum_{i=0}^{\mathtt{df}} X^2_i is chi-square distributed, denoted .. math:: Q \sim \chi^2_k. The probability density function of the chi-squared distribution is .. math:: p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}, where :math:`\Gamma` is the gamma function, .. math:: \Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt. References ---------- .. [1] NIST "Engineering Statistics Handbook" https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm Examples -------- >>> rng = np.random.default_rng() >>> rng.chisquare(2,4) array([ 1.89920014, 9.00867716, 3.13710533, 5.62318272]) # random The distribution of a chi-square random variable with 20 degrees of freedom looks as follows: >>> import matplotlib.pyplot as plt >>> import scipy.stats as stats >>> s = rng.chisquare(20, 10000) >>> count, bins, _ = plt.hist(s, 30, density=True) >>> x = np.linspace(0, 60, 1000) >>> plt.plot(x, stats.chi2.pdf(x, df=20)) >>> plt.xlim([0, 60]) >>> plt.show() noncentral_f(dfnum, dfden, nonc, size=None) Draw samples from the noncentral F distribution. Samples are drawn from an F distribution with specified parameters, `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of freedom in denominator), where both parameters > 1. `nonc` is the non-centrality parameter. Parameters ---------- dfnum : float or array_like of floats Numerator degrees of freedom, must be > 0. .. versionchanged:: 1.14.0 Earlier NumPy versions required dfnum > 1. dfden : float or array_like of floats Denominator degrees of freedom, must be > 0. nonc : float or array_like of floats Non-centrality parameter, the sum of the squares of the numerator means, must be >= 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``dfnum``, ``dfden``, and ``nonc`` are all scalars. Otherwise, ``np.broadcast(dfnum, dfden, nonc).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized noncentral Fisher distribution. Notes ----- When calculating the power of an experiment (power = probability of rejecting the null hypothesis when a specific alternative is true) the non-central F statistic becomes important. When the null hypothesis is true, the F statistic follows a central F distribution. When the null hypothesis is not true, then it follows a non-central F statistic. References ---------- .. [1] Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NoncentralF-Distribution.html .. [2] Wikipedia, "Noncentral F-distribution", https://en.wikipedia.org/wiki/Noncentral_F-distribution Examples -------- In a study, testing for a specific alternative to the null hypothesis requires use of the Noncentral F distribution. We need to calculate the area in the tail of the distribution that exceeds the value of the F distribution for the null hypothesis. We'll plot the two probability distributions for comparison. >>> rng = np.random.default_rng() >>> dfnum = 3 # between group deg of freedom >>> dfden = 20 # within groups degrees of freedom >>> nonc = 3.0 >>> nc_vals = rng.noncentral_f(dfnum, dfden, nonc, 1000000) >>> NF = np.histogram(nc_vals, bins=50, density=True) >>> c_vals = rng.f(dfnum, dfden, 1000000) >>> F = np.histogram(c_vals, bins=50, density=True) >>> import matplotlib.pyplot as plt >>> plt.plot(F[1][1:], F[0]) >>> plt.plot(NF[1][1:], NF[0]) >>> plt.show() f(dfnum, dfden, size=None) Draw samples from an F distribution. Samples are drawn from an F distribution with specified parameters, `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of freedom in denominator), where both parameters must be greater than zero. The random variate of the F distribution (also known as the Fisher distribution) is a continuous probability distribution that arises in ANOVA tests, and is the ratio of two chi-square variates. Parameters ---------- dfnum : float or array_like of floats Degrees of freedom in numerator, must be > 0. dfden : float or array_like of float Degrees of freedom in denominator, must be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``dfnum`` and ``dfden`` are both scalars. Otherwise, ``np.broadcast(dfnum, dfden).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Fisher distribution. See Also -------- scipy.stats.f : probability density function, distribution or cumulative density function, etc. Notes ----- The F statistic is used to compare in-group variances to between-group variances. Calculating the distribution depends on the sampling, and so it is a function of the respective degrees of freedom in the problem. The variable `dfnum` is the number of samples minus one, the between-groups degrees of freedom, while `dfden` is the within-groups degrees of freedom, the sum of the number of samples in each group minus the number of groups. References ---------- .. [1] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill, Fifth Edition, 2002. .. [2] Wikipedia, "F-distribution", https://en.wikipedia.org/wiki/F-distribution Examples -------- An example from Glantz[1], pp 47-40: Two groups, children of diabetics (25 people) and children from people without diabetes (25 controls). Fasting blood glucose was measured, case group had a mean value of 86.1, controls had a mean value of 82.2. Standard deviations were 2.09 and 2.49 respectively. Are these data consistent with the null hypothesis that the parents diabetic status does not affect their children's blood glucose levels? Calculating the F statistic from the data gives a value of 36.01. Draw samples from the distribution: >>> dfnum = 1. # between group degrees of freedom >>> dfden = 48. # within groups degrees of freedom >>> rng = np.random.default_rng() >>> s = rng.f(dfnum, dfden, 1000) The lower bound for the top 1% of the samples is : >>> np.sort(s)[-10] 7.61988120985 # random So there is about a 1% chance that the F statistic will exceed 7.62, the measured value is 36, so the null hypothesis is rejected at the 1% level. The corresponding probability density function for ``n = 20`` and ``m = 20`` is: >>> import matplotlib.pyplot as plt >>> from scipy import stats # doctest: +SKIP >>> dfnum, dfden, size = 20, 20, 10000 >>> s = rng.f(dfnum=dfnum, dfden=dfden, size=size) >>> bins, density, _ = plt.hist(s, 30, density=True) >>> x = np.linspace(0, 5, 1000) >>> plt.plot(x, stats.f.pdf(x, dfnum, dfden)) >>> plt.xlim([0, 5]) >>> plt.show() gamma(shape, scale=1.0, size=None) Draw samples from a Gamma distribution. Samples are drawn from a Gamma distribution with specified parameters, `shape` (sometimes designated "k") and `scale` (sometimes designated "theta"), where both parameters are > 0. Parameters ---------- shape : float or array_like of floats The shape of the gamma distribution. Must be non-negative. scale : float or array_like of floats, optional The scale of the gamma distribution. Must be non-negative. Default is equal to 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``shape`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(shape, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized gamma distribution. See Also -------- scipy.stats.gamma : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Gamma distribution is .. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)}, where :math:`k` is the shape and :math:`\theta` the scale, and :math:`\Gamma` is the Gamma function. The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant. References ---------- .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GammaDistribution.html .. [2] Wikipedia, "Gamma distribution", https://en.wikipedia.org/wiki/Gamma_distribution Examples -------- Draw samples from the distribution: >>> shape, scale = 2., 2. # mean=4, std=2*sqrt(2) >>> rng = np.random.default_rng() >>> s = rng.gamma(shape, scale, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> import scipy.special as sps # doctest: +SKIP >>> count, bins, _ = plt.hist(s, 50, density=True) >>> y = bins**(shape-1)*(np.exp(-bins/scale) / # doctest: +SKIP ... (sps.gamma(shape)*scale**shape)) >>> plt.plot(bins, y, linewidth=2, color='r') # doctest: +SKIP >>> plt.show() standard_gamma(shape, size=None, dtype=np.float64, out=None) Draw samples from a standard Gamma distribution. Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated "k") and scale=1. Parameters ---------- shape : float or array_like of floats Parameter, must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``shape`` is a scalar. Otherwise, ``np.array(shape).size`` samples are drawn. dtype : dtype, optional Desired dtype of the result, only `float64` and `float32` are supported. Byteorder must be native. The default value is np.float64. out : ndarray, optional Alternative output array in which to place the result. If size is not None, it must have the same shape as the provided size and must match the type of the output values. Returns ------- out : ndarray or scalar Drawn samples from the parameterized standard gamma distribution. See Also -------- scipy.stats.gamma : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Gamma distribution is .. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)}, where :math:`k` is the shape and :math:`\theta` the scale, and :math:`\Gamma` is the Gamma function. The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant. References ---------- .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GammaDistribution.html .. [2] Wikipedia, "Gamma distribution", https://en.wikipedia.org/wiki/Gamma_distribution Examples -------- Draw samples from the distribution: >>> shape, scale = 2., 1. # mean and width >>> rng = np.random.default_rng() >>> s = rng.standard_gamma(shape, 1000000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> import scipy.special as sps # doctest: +SKIP >>> count, bins, _ = plt.hist(s, 50, density=True) >>> y = bins**(shape-1) * ((np.exp(-bins/scale))/ # doctest: +SKIP ... (sps.gamma(shape) * scale**shape)) >>> plt.plot(bins, y, linewidth=2, color='r') # doctest: +SKIP >>> plt.show() normal(loc=0.0, scale=1.0, size=None) Draw random samples from a normal (Gaussian) distribution. The probability density function of the normal distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently [2]_, is often called the bell curve because of its characteristic shape (see the example below). The normal distributions occurs often in nature. For example, it describes the commonly occurring distribution of samples influenced by a large number of tiny, random disturbances, each with its own unique distribution [2]_. Parameters ---------- loc : float or array_like of floats Mean ("centre") of the distribution. scale : float or array_like of floats Standard deviation (spread or "width") of the distribution. Must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized normal distribution. See Also -------- scipy.stats.norm : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Gaussian distribution is .. math:: p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }} e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} }, where :math:`\mu` is the mean and :math:`\sigma` the standard deviation. The square of the standard deviation, :math:`\sigma^2`, is called the variance. The function has its peak at the mean, and its "spread" increases with the standard deviation (the function reaches 0.607 times its maximum at :math:`x + \sigma` and :math:`x - \sigma` [2]_). This implies that :meth:`normal` is more likely to return samples lying close to the mean, rather than those far away. References ---------- .. [1] Wikipedia, "Normal distribution", https://en.wikipedia.org/wiki/Normal_distribution .. [2] P. R. Peebles Jr., "Central Limit Theorem" in "Probability, Random Variables and Random Signal Principles", 4th ed., 2001, pp. 51, 51, 125. Examples -------- Draw samples from the distribution: >>> mu, sigma = 0, 0.1 # mean and standard deviation >>> rng = np.random.default_rng() >>> s = rng.normal(mu, sigma, 1000) Verify the mean and the variance: >>> abs(mu - np.mean(s)) 0.0 # may vary >>> abs(sigma - np.std(s, ddof=1)) 0.0 # may vary Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, 30, density=True) >>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) * ... np.exp( - (bins - mu)**2 / (2 * sigma**2) ), ... linewidth=2, color='r') >>> plt.show() Two-by-four array of samples from the normal distribution with mean 3 and standard deviation 2.5: >>> rng = np.random.default_rng() >>> rng.normal(3, 2.5, size=(2, 4)) array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random standard_normal(size=None, dtype=np.float64, out=None) Draw samples from a standard Normal distribution (mean=0, stdev=1). Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. dtype : dtype, optional Desired dtype of the result, only `float64` and `float32` are supported. Byteorder must be native. The default value is np.float64. out : ndarray, optional Alternative output array in which to place the result. If size is not None, it must have the same shape as the provided size and must match the type of the output values. Returns ------- out : float or ndarray A floating-point array of shape ``size`` of drawn samples, or a single sample if ``size`` was not specified. See Also -------- normal : Equivalent function with additional ``loc`` and ``scale`` arguments for setting the mean and standard deviation. Notes ----- For random samples from the normal distribution with mean ``mu`` and standard deviation ``sigma``, use one of:: mu + sigma * rng.standard_normal(size=...) rng.normal(mu, sigma, size=...) Examples -------- >>> rng = np.random.default_rng() >>> rng.standard_normal() 2.1923875335537315 # random >>> s = rng.standard_normal(8000) >>> s array([ 0.6888893 , 0.78096262, -0.89086505, ..., 0.49876311, # random -0.38672696, -0.4685006 ]) # random >>> s.shape (8000,) >>> s = rng.standard_normal(size=(3, 4, 2)) >>> s.shape (3, 4, 2) Two-by-four array of samples from the normal distribution with mean 3 and standard deviation 2.5: >>> 3 + 2.5 * rng.standard_normal(size=(2, 4)) array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random uniform(low=0.0, high=1.0, size=None) Draw samples from a uniform distribution. Samples are uniformly distributed over the half-open interval ``[low, high)`` (includes low, but excludes high). In other words, any value within the given interval is equally likely to be drawn by `uniform`. Parameters ---------- low : float or array_like of floats, optional Lower boundary of the output interval. All values generated will be greater than or equal to low. The default value is 0. high : float or array_like of floats Upper boundary of the output interval. All values generated will be less than high. The high limit may be included in the returned array of floats due to floating-point rounding in the equation ``low + (high-low) * random_sample()``. high - low must be non-negative. The default value is 1.0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``low`` and ``high`` are both scalars. Otherwise, ``np.broadcast(low, high).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized uniform distribution. See Also -------- integers : Discrete uniform distribution, yielding integers. random : Floats uniformly distributed over ``[0, 1)``. Notes ----- The probability density function of the uniform distribution is .. math:: p(x) = \frac{1}{b - a} anywhere within the interval ``[a, b)``, and zero elsewhere. When ``high`` == ``low``, values of ``low`` will be returned. Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> s = rng.uniform(-1,0,1000) All values are within the given interval: >>> np.all(s >= -1) True >>> np.all(s < 0) True Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, 15, density=True) >>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r') >>> plt.show() choice(a, size=None, replace=True, p=None, axis=0, shuffle=True) Generates a random sample from a given array Parameters ---------- a : {array_like, int} If an ndarray, a random sample is generated from its elements. If an int, the random sample is generated from np.arange(a). size : {int, tuple[int]}, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn from the 1-d `a`. If `a` has more than one dimension, the `size` shape will be inserted into the `axis` dimension, so the output ``ndim`` will be ``a.ndim - 1 + len(size)``. Default is None, in which case a single value is returned. replace : bool, optional Whether the sample is with or without replacement. Default is True, meaning that a value of ``a`` can be selected multiple times. p : 1-D array_like, optional The probabilities associated with each entry in a. If not given, the sample assumes a uniform distribution over all entries in ``a``. axis : int, optional The axis along which the selection is performed. The default, 0, selects by row. shuffle : bool, optional Whether the sample is shuffled when sampling without replacement. Default is True, False provides a speedup. Returns ------- samples : single item or ndarray The generated random samples Raises ------ ValueError If a is an int and less than zero, if p is not 1-dimensional, if a is array-like with a size 0, if p is not a vector of probabilities, if a and p have different lengths, or if replace=False and the sample size is greater than the population size. See Also -------- integers, shuffle, permutation Notes ----- Setting user-specified probabilities through ``p`` uses a more general but less efficient sampler than the default. The general sampler produces a different sample than the optimized sampler even if each element of ``p`` is 1 / len(a). ``p`` must sum to 1 when cast to ``float64``. To ensure this, you may wish to normalize using ``p = p / np.sum(p, dtype=float)``. Examples -------- Generate a uniform random sample from np.arange(5) of size 3: >>> rng = np.random.default_rng() >>> rng.choice(5, 3) array([0, 3, 4]) # random >>> #This is equivalent to rng.integers(0,5,3) Generate a non-uniform random sample from np.arange(5) of size 3: >>> rng.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0]) array([3, 3, 0]) # random Generate a uniform random sample from np.arange(5) of size 3 without replacement: >>> rng.choice(5, 3, replace=False) array([3,1,0]) # random >>> #This is equivalent to rng.permutation(np.arange(5))[:3] Generate a uniform random sample from a 2-D array along the first axis (the default), without replacement: >>> rng.choice([[0, 1, 2], [3, 4, 5], [6, 7, 8]], 2, replace=False) array([[3, 4, 5], # random [0, 1, 2]]) Generate a non-uniform random sample from np.arange(5) of size 3 without replacement: >>> rng.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0]) array([2, 3, 0]) # random Any of the above can be repeated with an arbitrary array-like instead of just integers. For instance: >>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher'] >>> rng.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3]) array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], # random dtype='>> rng = np.random.default_rng() >>> rng.bytes(10) b'\xfeC\x9b\x86\x17\xf2\xa1\xafcp' # random integers(low, high=None, size=None, dtype=np.int64, endpoint=False) Return random integers from `low` (inclusive) to `high` (exclusive), or if endpoint=True, `low` (inclusive) to `high` (inclusive). Replaces `RandomState.randint` (with endpoint=False) and `RandomState.random_integers` (with endpoint=True) Return random integers from the "discrete uniform" distribution of the specified dtype. If `high` is None (the default), then results are from 0 to `low`. Parameters ---------- low : int or array-like of ints Lowest (signed) integers to be drawn from the distribution (unless ``high=None``, in which case this parameter is 0 and this value is used for `high`). high : int or array-like of ints, optional If provided, one above the largest (signed) integer to be drawn from the distribution (see above for behavior if ``high=None``). If array-like, must contain integer values size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. dtype : dtype, optional Desired dtype of the result. Byteorder must be native. The default value is np.int64. endpoint : bool, optional If true, sample from the interval [low, high] instead of the default [low, high) Defaults to False Returns ------- out : int or ndarray of ints `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. Notes ----- When using broadcasting with uint64 dtypes, the maximum value (2**64) cannot be represented as a standard integer type. The high array (or low if high is None) must have object dtype, e.g., array([2**64]). Examples -------- >>> rng = np.random.default_rng() >>> rng.integers(2, size=10) array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) # random >>> rng.integers(1, size=10) array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]) Generate a 2 x 4 array of ints between 0 and 4, inclusive: >>> rng.integers(5, size=(2, 4)) array([[4, 0, 2, 1], [3, 2, 2, 0]]) # random Generate a 1 x 3 array with 3 different upper bounds >>> rng.integers(1, [3, 5, 10]) array([2, 2, 9]) # random Generate a 1 by 3 array with 3 different lower bounds >>> rng.integers([1, 5, 7], 10) array([9, 8, 7]) # random Generate a 2 by 4 array using broadcasting with dtype of uint8 >>> rng.integers([1, 3, 5, 7], [[10], [20]], dtype=np.uint8) array([[ 8, 6, 9, 7], [ 1, 16, 9, 12]], dtype=uint8) # random References ---------- .. [1] Daniel Lemire., "Fast Random Integer Generation in an Interval", ACM Transactions on Modeling and Computer Simulation 29 (1), 2019, https://arxiv.org/abs/1805.10941. standard_exponential(size=None, dtype=np.float64, method='zig', out=None) Draw samples from the standard exponential distribution. `standard_exponential` is identical to the exponential distribution with a scale parameter of 1. Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. dtype : dtype, optional Desired dtype of the result, only `float64` and `float32` are supported. Byteorder must be native. The default value is np.float64. method : str, optional Either 'inv' or 'zig'. 'inv' uses the default inverse CDF method. 'zig' uses the much faster Ziggurat method of Marsaglia and Tsang. out : ndarray, optional Alternative output array in which to place the result. If size is not None, it must have the same shape as the provided size and must match the type of the output values. Returns ------- out : float or ndarray Drawn samples. Examples -------- Output a 3x8000 array: >>> rng = np.random.default_rng() >>> n = rng.standard_exponential((3, 8000)) exponential(scale=1.0, size=None) Draw samples from an exponential distribution. Its probability density function is .. math:: f(x; \frac{1}{\beta}) = \frac{1}{\beta} \exp(-\frac{x}{\beta}), for ``x > 0`` and 0 elsewhere. :math:`\beta` is the scale parameter, which is the inverse of the rate parameter :math:`\lambda = 1/\beta`. The rate parameter is an alternative, widely used parameterization of the exponential distribution [3]_. The exponential distribution is a continuous analogue of the geometric distribution. It describes many common situations, such as the size of raindrops measured over many rainstorms [1]_, or the time between page requests to Wikipedia [2]_. Parameters ---------- scale : float or array_like of floats The scale parameter, :math:`\beta = 1/\lambda`. Must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``scale`` is a scalar. Otherwise, ``np.array(scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized exponential distribution. Examples -------- Assume a company has 10000 customer support agents and the time between customer calls is exponentially distributed and that the average time between customer calls is 4 minutes. >>> scale, size = 4, 10000 >>> rng = np.random.default_rng() >>> time_between_calls = rng.exponential(scale=scale, size=size) What is the probability that a customer will call in the next 4 to 5 minutes? >>> x = ((time_between_calls < 5).sum())/size >>> y = ((time_between_calls < 4).sum())/size >>> x - y 0.08 # may vary The corresponding distribution can be visualized as follows: >>> import matplotlib.pyplot as plt >>> scale, size = 4, 10000 >>> rng = np.random.default_rng() >>> sample = rng.exponential(scale=scale, size=size) >>> count, bins, _ = plt.hist(sample, 30, density=True) >>> plt.plot(bins, scale**(-1)*np.exp(-scale**-1*bins), linewidth=2, color='r') >>> plt.show() References ---------- .. [1] Peyton Z. Peebles Jr., "Probability, Random Variables and Random Signal Principles", 4th ed, 2001, p. 57. .. [2] Wikipedia, "Poisson process", https://en.wikipedia.org/wiki/Poisson_process .. [3] Wikipedia, "Exponential distribution", https://en.wikipedia.org/wiki/Exponential_distribution beta(a, b, size=None) Draw samples from a Beta distribution. The Beta distribution is a special case of the Dirichlet distribution, and is related to the Gamma distribution. It has the probability distribution function .. math:: f(x; a,b) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1} (1 - x)^{\beta - 1}, where the normalization, B, is the beta function, .. math:: B(\alpha, \beta) = \int_0^1 t^{\alpha - 1} (1 - t)^{\beta - 1} dt. It is often seen in Bayesian inference and order statistics. Parameters ---------- a : float or array_like of floats Alpha, positive (>0). b : float or array_like of floats Beta, positive (>0). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` and ``b`` are both scalars. Otherwise, ``np.broadcast(a, b).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized beta distribution. Examples -------- The beta distribution has mean a/(a+b). If ``a == b`` and both are > 1, the distribution is symmetric with mean 0.5. >>> rng = np.random.default_rng() >>> a, b, size = 2.0, 2.0, 10000 >>> sample = rng.beta(a=a, b=b, size=size) >>> np.mean(sample) 0.5047328775385895 # may vary Otherwise the distribution is skewed left or right according to whether ``a`` or ``b`` is greater. The distribution is mirror symmetric. See for example: >>> a, b, size = 2, 7, 10000 >>> sample_left = rng.beta(a=a, b=b, size=size) >>> sample_right = rng.beta(a=b, b=a, size=size) >>> m_left, m_right = np.mean(sample_left), np.mean(sample_right) >>> print(m_left, m_right) 0.2238596793678923 0.7774613834041182 # may vary >>> print(m_left - a/(a+b)) 0.001637457145670096 # may vary >>> print(m_right - b/(a+b)) -0.0003163943736596009 # may vary Display the histogram of the two samples: >>> import matplotlib.pyplot as plt >>> plt.hist([sample_left, sample_right], ... 50, density=True, histtype='bar') >>> plt.show() References ---------- .. [1] Wikipedia, "Beta distribution", https://en.wikipedia.org/wiki/Beta_distribution random(size=None, dtype=np.float64, out=None) Return random floats in the half-open interval [0.0, 1.0). Results are from the "continuous uniform" distribution over the stated interval. To sample :math:`Unif[a, b), b > a` use `uniform` or multiply the output of `random` by ``(b - a)`` and add ``a``:: (b - a) * random() + a Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. dtype : dtype, optional Desired dtype of the result, only `float64` and `float32` are supported. Byteorder must be native. The default value is np.float64. out : ndarray, optional Alternative output array in which to place the result. If size is not None, it must have the same shape as the provided size and must match the type of the output values. Returns ------- out : float or ndarray of floats Array of random floats of shape `size` (unless ``size=None``, in which case a single float is returned). See Also -------- uniform : Draw samples from the parameterized uniform distribution. Examples -------- >>> rng = np.random.default_rng() >>> rng.random() 0.47108547995356098 # random >>> type(rng.random()) >>> rng.random((5,)) array([ 0.30220482, 0.86820401, 0.1654503 , 0.11659149, 0.54323428]) # random Three-by-two array of random numbers from [-5, 0): >>> 5 * rng.random((3, 2)) - 5 array([[-3.99149989, -0.52338984], # random [-2.99091858, -0.79479508], [-1.23204345, -1.75224494]]) spawn(n_children) Create new independent child generators. See :ref:`seedsequence-spawn` for additional notes on spawning children. .. versionadded:: 1.25.0 Parameters ---------- n_children : int Returns ------- child_generators : list of Generators Raises ------ TypeError When the underlying SeedSequence does not implement spawning. See Also -------- random.BitGenerator.spawn, random.SeedSequence.spawn : Equivalent method on the bit generator and seed sequence. bit_generator : The bit generator instance used by the generator. Examples -------- Starting from a seeded default generator: >>> # High quality entropy created with: f"0x{secrets.randbits(128):x}" >>> entropy = 0x3034c61a9ae04ff8cb62ab8ec2c4b501 >>> rng = np.random.default_rng(entropy) Create two new generators for example for parallel execution: >>> child_rng1, child_rng2 = rng.spawn(2) Drawn numbers from each are independent but derived from the initial seeding entropy: >>> rng.uniform(), child_rng1.uniform(), child_rng2.uniform() (0.19029263503854454, 0.9475673279178444, 0.4702687338396767) It is safe to spawn additional children from the original ``rng`` or the children: >>> more_child_rngs = rng.spawn(20) >>> nested_spawn = child_rng1.spawn(20) Unsupported dtype %r for standard_gammaUnsupported dtype %r for standard_normalGenerator.multivariate_hypergeometric (line 4210)unable to allocate shape and strides.sum(colors) must not exceed the maximum value of a 64 bit signed integer (%d)pvals must have at least 1 dimension and the last dimension of pvals must be greater than 0.numpy._core.umath failed to importnumpy._core.multiarray failed to importno default __reduce__ due to non-trivial __cinit__negative dimensions are not allowedn too large or p too small, see Generator.negative_binomial Notesmethod must be one of {'eigh', 'svd', 'cholesky'}method must be "count" or "marginals".memory allocation failed in permutedmean and cov must not be complexmean and cov must have same lengthhigh - low range exceeds valid boundsgot differing extents in dimension covariance is not symmetric positive-semidefinite.cov must be 2 dimensional and squarecolors must be a one-dimensional sequence of nonnegative integers not exceeding %d.check_valid must equal 'warn', 'raise', or 'ignore'both ngood and nbad must be less than %da must be a sequence or an integer, not a must be a positive integer unless no samples are takena cannot be empty unless no samples are takenWhen method is "marginals", sum(colors) must be less than 1000000000.Unsupported dtype %r for standard_exponentialUnsupported dtype %r for integersUnable to convert item to objectProviding a dtype with a non-native byteorder is not supported. If you require platform-independent byteorder, call byteswap when required.Probabilities do not sum to 1. See Notes section of docstring for more information.Probabilities are not non-negativeOut of bounds on buffer access (axis Invalid mode, expected 'c' or 'fortran', got Invalid bit generator. The bit generator must be instantiated.Insufficient memory for multivariate_hypergeometric with method='count' and sum(colors)=%dIndirect dimensions not supportedIncompatible checksums (0x%x vs (0x82a3537, 0x6ae9995, 0xb068931) = (name))Generator.standard_normal (line 1116)Generator.standard_gamma (line 1293)Generator.standard_exponential (line 525)Generator.standard_cauchy (line 1805)Generator.noncentral_f (line 1566)Generator.noncentral_chisquare (line 1725)Generator.negative_binomial (line 3131)Generator.multivariate_normal (line 3721)Generator.multivariate_hypergeometricGenerator.hypergeometric (line 3494)Fewer non-zero entries in p than sizeEmpty shape tuple for cython.arrayConstruct a new Generator with the default BitGenerator (PCG64). Parameters ---------- seed : {None, int, array_like[ints], SeedSequence, BitGenerator, Generator}, optional A seed to initialize the `BitGenerator`. If None, then fresh, unpredictable entropy will be pulled from the OS. If an ``int`` or ``array_like[ints]`` is passed, then all values must be non-negative and will be passed to `SeedSequence` to derive the initial `BitGenerator` state. One may also pass in a `SeedSequence` instance. Additionally, when passed a `BitGenerator`, it will be wrapped by `Generator`. If passed a `Generator`, it will be returned unaltered. Returns ------- Generator The initialized generator object. Notes ----- If ``seed`` is not a `BitGenerator` or a `Generator`, a new `BitGenerator` is instantiated. This function does not manage a default global instance. See :ref:`seeding_and_entropy` for more information about seeding. Examples -------- `default_rng` is the recommended constructor for the random number class `Generator`. Here are several ways we can construct a random number generator using `default_rng` and the `Generator` class. Here we use `default_rng` to generate a random float: >>> import numpy as np >>> rng = np.random.default_rng(12345) >>> print(rng) Generator(PCG64) >>> rfloat = rng.random() >>> rfloat 0.22733602246716966 >>> type(rfloat) Here we use `default_rng` to generate 3 random integers between 0 (inclusive) and 10 (exclusive): >>> import numpy as np >>> rng = np.random.default_rng(12345) >>> rints = rng.integers(low=0, high=10, size=3) >>> rints array([6, 2, 7]) >>> type(rints[0]) Here we specify a seed so that we have reproducible results: >>> import numpy as np >>> rng = np.random.default_rng(seed=42) >>> print(rng) Generator(PCG64) >>> arr1 = rng.random((3, 3)) >>> arr1 array([[0.77395605, 0.43887844, 0.85859792], [0.69736803, 0.09417735, 0.97562235], [0.7611397 , 0.78606431, 0.12811363]]) If we exit and restart our Python interpreter, we'll see that we generate the same random numbers again: >>> import numpy as np >>> rng = np.random.default_rng(seed=42) >>> arr2 = rng.random((3, 3)) >>> arr2 array([[0.77395605, 0.43887844, 0.85859792], [0.69736803, 0.09417735, 0.97562235], [0.7611397 , 0.78606431, 0.12811363]]) Cannot transpose memoryview with indirect dimensionsCannot take a larger sample than population when replace is FalseCannot create writable memory view from read-only memoryviewCannot assign to read-only memoryviewCan only create a buffer that is contiguous in memory.Buffer view does not expose stridesAll dimensions preceding dimension %d must be indexed and not sliced zipf(a, size=None) Draw samples from a Zipf distribution. Samples are drawn from a Zipf distribution with specified parameter `a` > 1. The Zipf distribution (also known as the zeta distribution) is a discrete probability distribution that satisfies Zipf's law: the frequency of an item is inversely proportional to its rank in a frequency table. Parameters ---------- a : float or array_like of floats Distribution parameter. Must be greater than 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Zipf distribution. See Also -------- scipy.stats.zipf : probability density function, distribution, or cumulative density function, etc. Notes ----- The probability density for the Zipf distribution is .. math:: p(k) = \frac{k^{-a}}{\zeta(a)}, for integers :math:`k \geq 1`, where :math:`\zeta` is the Riemann Zeta function. It is named for the American linguist George Kingsley Zipf, who noted that the frequency of any word in a sample of a language is inversely proportional to its rank in the frequency table. References ---------- .. [1] Zipf, G. K., "Selected Studies of the Principle of Relative Frequency in Language," Cambridge, MA: Harvard Univ. Press, 1932. Examples -------- Draw samples from the distribution: >>> a = 4.0 >>> n = 20000 >>> rng = np.random.default_rng() >>> s = rng.zipf(a, size=n) Display the histogram of the samples, along with the expected histogram based on the probability density function: >>> import matplotlib.pyplot as plt >>> from scipy.special import zeta # doctest: +SKIP `bincount` provides a fast histogram for small integers. >>> count = np.bincount(s) >>> k = np.arange(1, s.max() + 1) >>> plt.bar(k, count[1:], alpha=0.5, label='sample count') >>> plt.plot(k, n*(k**-a)/zeta(a), 'k.-', alpha=0.5, ... label='expected count') # doctest: +SKIP >>> plt.semilogy() >>> plt.grid(alpha=0.4) >>> plt.legend() >>> plt.title(f'Zipf sample, a={a}, size={n}') >>> plt.show() weibull(a, size=None) Draw samples from a Weibull distribution. Draw samples from a 1-parameter Weibull distribution with the given shape parameter `a`. .. math:: X = (-ln(U))^{1/a} Here, U is drawn from the uniform distribution over (0,1]. The more common 2-parameter Weibull, including a scale parameter :math:`\lambda` is just :math:`X = \lambda(-ln(U))^{1/a}`. Parameters ---------- a : float or array_like of floats Shape parameter of the distribution. Must be nonnegative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Weibull distribution. See Also -------- scipy.stats.weibull_max scipy.stats.weibull_min scipy.stats.genextreme gumbel Notes ----- The Weibull (or Type III asymptotic extreme value distribution for smallest values, SEV Type III, or Rosin-Rammler distribution) is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. This class includes the Gumbel and Frechet distributions. The probability density for the Weibull distribution is .. math:: p(x) = \frac{a} {\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a}, where :math:`a` is the shape and :math:`\lambda` the scale. The function has its peak (the mode) at :math:`\lambda(\frac{a-1}{a})^{1/a}`. When ``a = 1``, the Weibull distribution reduces to the exponential distribution. References ---------- .. [1] Waloddi Weibull, Royal Technical University, Stockholm, 1939 "A Statistical Theory Of The Strength Of Materials", Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939, Generalstabens Litografiska Anstalts Forlag, Stockholm. .. [2] Waloddi Weibull, "A Statistical Distribution Function of Wide Applicability", Journal Of Applied Mechanics ASME Paper 1951. .. [3] Wikipedia, "Weibull distribution", https://en.wikipedia.org/wiki/Weibull_distribution Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> a = 5. # shape >>> s = rng.weibull(a, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> def weibull(x, n, a): ... return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a) >>> count, bins, _ = plt.hist(rng.weibull(5., 1000)) >>> x = np.linspace(0, 2, 1000) >>> bin_spacing = np.mean(np.diff(bins)) >>> plt.plot(x, weibull(x, 1., 5.) * bin_spacing * s.size, label='Weibull PDF') >>> plt.legend() >>> plt.show() vonmises(mu, kappa, size=None) Draw samples from a von Mises distribution. Samples are drawn from a von Mises distribution with specified mode (mu) and dispersion (kappa), on the interval [-pi, pi]. The von Mises distribution (also known as the circular normal distribution) is a continuous probability distribution on the unit circle. It may be thought of as the circular analogue of the normal distribution. Parameters ---------- mu : float or array_like of floats Mode ("center") of the distribution. kappa : float or array_like of floats Dispersion of the distribution, has to be >=0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``mu`` and ``kappa`` are both scalars. Otherwise, ``np.broadcast(mu, kappa).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized von Mises distribution. See Also -------- scipy.stats.vonmises : probability density function, distribution, or cumulative density function, etc. Notes ----- The probability density for the von Mises distribution is .. math:: p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}, where :math:`\mu` is the mode and :math:`\kappa` the dispersion, and :math:`I_0(\kappa)` is the modified Bessel function of order 0. The von Mises is named for Richard Edler von Mises, who was born in Austria-Hungary, in what is now the Ukraine. He fled to the United States in 1939 and became a professor at Harvard. He worked in probability theory, aerodynamics, fluid mechanics, and philosophy of science. References ---------- .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing," New York: Dover, 1972. .. [2] von Mises, R., "Mathematical Theory of Probability and Statistics", New York: Academic Press, 1964. Examples -------- Draw samples from the distribution: >>> mu, kappa = 0.0, 4.0 # mean and dispersion >>> rng = np.random.default_rng() >>> s = rng.vonmises(mu, kappa, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> from scipy.special import i0 # doctest: +SKIP >>> plt.hist(s, 50, density=True) >>> x = np.linspace(-np.pi, np.pi, num=51) >>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa)) # doctest: +SKIP >>> plt.plot(x, y, linewidth=2, color='r') # doctest: +SKIP >>> plt.show() uniform(low=0.0, high=1.0, size=None) Draw samples from a uniform distribution. Samples are uniformly distributed over the half-open interval ``[low, high)`` (includes low, but excludes high). In other words, any value within the given interval is equally likely to be drawn by `uniform`. Parameters ---------- low : float or array_like of floats, optional Lower boundary of the output interval. All values generated will be greater than or equal to low. The default value is 0. high : float or array_like of floats Upper boundary of the output interval. All values generated will be less than high. The high limit may be included in the returned array of floats due to floating-point rounding in the equation ``low + (high-low) * random_sample()``. high - low must be non-negative. The default value is 1.0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``low`` and ``high`` are both scalars. Otherwise, ``np.broadcast(low, high).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized uniform distribution. See Also -------- integers : Discrete uniform distribution, yielding integers. random : Floats uniformly distributed over ``[0, 1)``. Notes ----- The probability density function of the uniform distribution is .. math:: p(x) = \frac{1}{b - a} anywhere within the interval ``[a, b)``, and zero elsewhere. When ``high`` == ``low``, values of ``low`` will be returned. Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> s = rng.uniform(-1,0,1000) All values are within the given interval: >>> np.all(s >= -1) True >>> np.all(s < 0) True Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, 15, density=True) >>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r') >>> plt.show() triangular(left, mode, right, size=None) Draw samples from the triangular distribution over the interval ``[left, right]``. The triangular distribution is a continuous probability distribution with lower limit left, peak at mode, and upper limit right. Unlike the other distributions, these parameters directly define the shape of the pdf. Parameters ---------- left : float or array_like of floats Lower limit. mode : float or array_like of floats The value where the peak of the distribution occurs. The value must fulfill the condition ``left <= mode <= right``. right : float or array_like of floats Upper limit, must be larger than `left`. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``left``, ``mode``, and ``right`` are all scalars. Otherwise, ``np.broadcast(left, mode, right).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized triangular distribution. Notes ----- The probability density function for the triangular distribution is .. math:: P(x;l, m, r) = \begin{cases} \frac{2(x-l)}{(r-l)(m-l)}& \text{for $l \leq x \leq m$},\\ \frac{2(r-x)}{(r-l)(r-m)}& \text{for $m \leq x \leq r$},\\ 0& \text{otherwise}. \end{cases} The triangular distribution is often used in ill-defined problems where the underlying distribution is not known, but some knowledge of the limits and mode exists. Often it is used in simulations. References ---------- .. [1] Wikipedia, "Triangular distribution" https://en.wikipedia.org/wiki/Triangular_distribution Examples -------- Draw values from the distribution and plot the histogram: >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() >>> h = plt.hist(rng.triangular(-3, 0, 8, 100000), bins=200, ... density=True) >>> plt.show() standard_t(df, size=None) Draw samples from a standard Student's t distribution with `df` degrees of freedom. A special case of the hyperbolic distribution. As `df` gets large, the result resembles that of the standard normal distribution (`standard_normal`). Parameters ---------- df : float or array_like of floats Degrees of freedom, must be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``df`` is a scalar. Otherwise, ``np.array(df).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized standard Student's t distribution. Notes ----- The probability density function for the t distribution is .. math:: P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df} \Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2} The t test is based on an assumption that the data come from a Normal distribution. The t test provides a way to test whether the sample mean (that is the mean calculated from the data) is a good estimate of the true mean. The derivation of the t-distribution was first published in 1908 by William Gosset while working for the Guinness Brewery in Dublin. Due to proprietary issues, he had to publish under a pseudonym, and so he used the name Student. References ---------- .. [1] Dalgaard, Peter, "Introductory Statistics With R", Springer, 2002. .. [2] Wikipedia, "Student's t-distribution" https://en.wikipedia.org/wiki/Student's_t-distribution Examples -------- From Dalgaard page 83 [1]_, suppose the daily energy intake for 11 women in kilojoules (kJ) is: >>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \ ... 7515, 8230, 8770]) Does their energy intake deviate systematically from the recommended value of 7725 kJ? Our null hypothesis will be the absence of deviation, and the alternate hypothesis will be the presence of an effect that could be either positive or negative, hence making our test 2-tailed. Because we are estimating the mean and we have N=11 values in our sample, we have N-1=10 degrees of freedom. We set our significance level to 95% and compute the t statistic using the empirical mean and empirical standard deviation of our intake. We use a ddof of 1 to base the computation of our empirical standard deviation on an unbiased estimate of the variance (note: the final estimate is not unbiased due to the concave nature of the square root). >>> np.mean(intake) 6753.636363636364 >>> intake.std(ddof=1) 1142.1232221373727 >>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake))) >>> t -2.8207540608310198 We draw 1000000 samples from Student's t distribution with the adequate degrees of freedom. >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() >>> s = rng.standard_t(10, size=1000000) >>> h = plt.hist(s, bins=100, density=True) Does our t statistic land in one of the two critical regions found at both tails of the distribution? >>> np.sum(np.abs(t) < np.abs(s)) / float(len(s)) 0.018318 #random < 0.05, statistic is in critical region The probability value for this 2-tailed test is about 1.83%, which is lower than the 5% pre-determined significance threshold. Therefore, the probability of observing values as extreme as our intake conditionally on the null hypothesis being true is too low, and we reject the null hypothesis of no deviation. standard_normal(size=None, dtype=np.float64, out=None) Draw samples from a standard Normal distribution (mean=0, stdev=1). Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. dtype : dtype, optional Desired dtype of the result, only `float64` and `float32` are supported. Byteorder must be native. The default value is np.float64. out : ndarray, optional Alternative output array in which to place the result. If size is not None, it must have the same shape as the provided size and must match the type of the output values. Returns ------- out : float or ndarray A floating-point array of shape ``size`` of drawn samples, or a single sample if ``size`` was not specified. See Also -------- normal : Equivalent function with additional ``loc`` and ``scale`` arguments for setting the mean and standard deviation. Notes ----- For random samples from the normal distribution with mean ``mu`` and standard deviation ``sigma``, use one of:: mu + sigma * rng.standard_normal(size=...) rng.normal(mu, sigma, size=...) Examples -------- >>> rng = np.random.default_rng() >>> rng.standard_normal() 2.1923875335537315 # random >>> s = rng.standard_normal(8000) >>> s array([ 0.6888893 , 0.78096262, -0.89086505, ..., 0.49876311, # random -0.38672696, -0.4685006 ]) # random >>> s.shape (8000,) >>> s = rng.standard_normal(size=(3, 4, 2)) >>> s.shape (3, 4, 2) Two-by-four array of samples from the normal distribution with mean 3 and standard deviation 2.5: >>> 3 + 2.5 * rng.standard_normal(size=(2, 4)) array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random spawn(n_children) Create new independent child generators. See :ref:`seedsequence-spawn` for additional notes on spawning children. .. versionadded:: 1.25.0 Parameters ---------- n_children : int Returns ------- child_generators : list of Generators Raises ------ TypeError When the underlying SeedSequence does not implement spawning. See Also -------- random.BitGenerator.spawn, random.SeedSequence.spawn : Equivalent method on the bit generator and seed sequence. bit_generator : The bit generator instance used by the generator. Examples -------- Starting from a seeded default generator: >>> # High quality entropy created with: f"0x{secrets.randbits(128):x}" >>> entropy = 0x3034c61a9ae04ff8cb62ab8ec2c4b501 >>> rng = np.random.default_rng(entropy) Create two new generators for example for parallel execution: >>> child_rng1, child_rng2 = rng.spawn(2) Drawn numbers from each are independent but derived from the initial seeding entropy: >>> rng.uniform(), child_rng1.uniform(), child_rng2.uniform() (0.19029263503854454, 0.9475673279178444, 0.4702687338396767) It is safe to spawn additional children from the original ``rng`` or the children: >>> more_child_rngs = rng.spawn(20) >>> nested_spawn = child_rng1.spawn(20) shuffle(x, axis=0) Modify an array or sequence in-place by shuffling its contents. The order of sub-arrays is changed but their contents remains the same. Parameters ---------- x : ndarray or MutableSequence The array, list or mutable sequence to be shuffled. axis : int, optional The axis which `x` is shuffled along. Default is 0. It is only supported on `ndarray` objects. Returns ------- None See Also -------- permuted permutation Notes ----- An important distinction between methods ``shuffle`` and ``permuted`` is how they both treat the ``axis`` parameter which can be found at :ref:`generator-handling-axis-parameter`. Examples -------- >>> rng = np.random.default_rng() >>> arr = np.arange(10) >>> arr array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]) >>> rng.shuffle(arr) >>> arr array([2, 0, 7, 5, 1, 4, 8, 9, 3, 6]) # random >>> arr = np.arange(9).reshape((3, 3)) >>> arr array([[0, 1, 2], [3, 4, 5], [6, 7, 8]]) >>> rng.shuffle(arr) >>> arr array([[3, 4, 5], # random [6, 7, 8], [0, 1, 2]]) >>> arr = np.arange(9).reshape((3, 3)) >>> arr array([[0, 1, 2], [3, 4, 5], [6, 7, 8]]) >>> rng.shuffle(arr, axis=1) >>> arr array([[2, 0, 1], # random [5, 3, 4], [8, 6, 7]]) rayleigh(scale=1.0, size=None) Draw samples from a Rayleigh distribution. The :math:`\chi` and Weibull distributions are generalizations of the Rayleigh. Parameters ---------- scale : float or array_like of floats, optional Scale, also equals the mode. Must be non-negative. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``scale`` is a scalar. Otherwise, ``np.array(scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Rayleigh distribution. Notes ----- The probability density function for the Rayleigh distribution is .. math:: P(x;scale) = \frac{x}{scale^2}e^{\frac{-x^2}{2 \cdotp scale^2}} The Rayleigh distribution would arise, for example, if the East and North components of the wind velocity had identical zero-mean Gaussian distributions. Then the wind speed would have a Rayleigh distribution. References ---------- .. [1] Brighton Webs Ltd., "Rayleigh Distribution," https://web.archive.org/web/20090514091424/http://brighton-webs.co.uk:80/distributions/rayleigh.asp .. [2] Wikipedia, "Rayleigh distribution" https://en.wikipedia.org/wiki/Rayleigh_distribution Examples -------- Draw values from the distribution and plot the histogram >>> from matplotlib.pyplot import hist >>> rng = np.random.default_rng() >>> values = hist(rng.rayleigh(3, 100000), bins=200, density=True) Wave heights tend to follow a Rayleigh distribution. If the mean wave height is 1 meter, what fraction of waves are likely to be larger than 3 meters? >>> meanvalue = 1 >>> modevalue = np.sqrt(2 / np.pi) * meanvalue >>> s = rng.rayleigh(modevalue, 1000000) The percentage of waves larger than 3 meters is: >>> 100.*sum(s>3)/1000000. 0.087300000000000003 # random random(size=None, dtype=np.float64, out=None) Return random floats in the half-open interval [0.0, 1.0). Results are from the "continuous uniform" distribution over the stated interval. To sample :math:`Unif[a, b), b > a` use `uniform` or multiply the output of `random` by ``(b - a)`` and add ``a``:: (b - a) * random() + a Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. dtype : dtype, optional Desired dtype of the result, only `float64` and `float32` are supported. Byteorder must be native. The default value is np.float64. out : ndarray, optional Alternative output array in which to place the result. If size is not None, it must have the same shape as the provided size and must match the type of the output values. Returns ------- out : float or ndarray of floats Array of random floats of shape `size` (unless ``size=None``, in which case a single float is returned). See Also -------- uniform : Draw samples from the parameterized uniform distribution. Examples -------- >>> rng = np.random.default_rng() >>> rng.random() 0.47108547995356098 # random >>> type(rng.random()) >>> rng.random((5,)) array([ 0.30220482, 0.86820401, 0.1654503 , 0.11659149, 0.54323428]) # random Three-by-two array of random numbers from [-5, 0): >>> 5 * rng.random((3, 2)) - 5 array([[-3.99149989, -0.52338984], # random [-2.99091858, -0.79479508], [-1.23204345, -1.75224494]]) power(a, size=None) Draws samples in [0, 1] from a power distribution with positive exponent a - 1. Also known as the power function distribution. Parameters ---------- a : float or array_like of floats Parameter of the distribution. Must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized power distribution. Raises ------ ValueError If a <= 0. Notes ----- The probability density function is .. math:: P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0. The power function distribution is just the inverse of the Pareto distribution. It may also be seen as a special case of the Beta distribution. It is used, for example, in modeling the over-reporting of insurance claims. References ---------- .. [1] Christian Kleiber, Samuel Kotz, "Statistical size distributions in economics and actuarial sciences", Wiley, 2003. .. [2] Heckert, N. A. and Filliben, James J. "NIST Handbook 148: Dataplot Reference Manual, Volume 2: Let Subcommands and Library Functions", National Institute of Standards and Technology Handbook Series, June 2003. https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> a = 5. # shape >>> samples = 1000 >>> s = rng.power(a, samples) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, bins=30) >>> x = np.linspace(0, 1, 100) >>> y = a*x**(a-1.) >>> normed_y = samples*np.diff(bins)[0]*y >>> plt.plot(x, normed_y) >>> plt.show() Compare the power function distribution to the inverse of the Pareto. >>> from scipy import stats # doctest: +SKIP >>> rvs = rng.power(5, 1000000) >>> rvsp = rng.pareto(5, 1000000) >>> xx = np.linspace(0,1,100) >>> powpdf = stats.powerlaw.pdf(xx,5) # doctest: +SKIP >>> plt.figure() >>> plt.hist(rvs, bins=50, density=True) >>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP >>> plt.title('power(5)') >>> plt.figure() >>> plt.hist(1./(1.+rvsp), bins=50, density=True) >>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP >>> plt.title('inverse of 1 + Generator.pareto(5)') >>> plt.figure() >>> plt.hist(1./(1.+rvsp), bins=50, density=True) >>> plt.plot(xx,powpdf,'r-') # doctest: +SKIP >>> plt.title('inverse of stats.pareto(5)') permuted(x, axis=None, out=None) Randomly permute `x` along axis `axis`. Unlike `shuffle`, each slice along the given axis is shuffled independently of the others. Parameters ---------- x : array_like, at least one-dimensional Array to be shuffled. axis : int, optional Slices of `x` in this axis are shuffled. Each slice is shuffled independently of the others. If `axis` is None, the flattened array is shuffled. out : ndarray, optional If given, this is the destination of the shuffled array. If `out` is None, a shuffled copy of the array is returned. Returns ------- ndarray If `out` is None, a shuffled copy of `x` is returned. Otherwise, the shuffled array is stored in `out`, and `out` is returned See Also -------- shuffle permutation Notes ----- An important distinction between methods ``shuffle`` and ``permuted`` is how they both treat the ``axis`` parameter which can be found at :ref:`generator-handling-axis-parameter`. Examples -------- Create a `numpy.random.Generator` instance: >>> rng = np.random.default_rng() Create a test array: >>> x = np.arange(24).reshape(3, 8) >>> x array([[ 0, 1, 2, 3, 4, 5, 6, 7], [ 8, 9, 10, 11, 12, 13, 14, 15], [16, 17, 18, 19, 20, 21, 22, 23]]) Shuffle the rows of `x`: >>> y = rng.permuted(x, axis=1) >>> y array([[ 4, 3, 6, 7, 1, 2, 5, 0], # random [15, 10, 14, 9, 12, 11, 8, 13], [17, 16, 20, 21, 18, 22, 23, 19]]) `x` has not been modified: >>> x array([[ 0, 1, 2, 3, 4, 5, 6, 7], [ 8, 9, 10, 11, 12, 13, 14, 15], [16, 17, 18, 19, 20, 21, 22, 23]]) To shuffle the rows of `x` in-place, pass `x` as the `out` parameter: >>> y = rng.permuted(x, axis=1, out=x) >>> x array([[ 3, 0, 4, 7, 1, 6, 2, 5], # random [ 8, 14, 13, 9, 12, 11, 15, 10], [17, 18, 16, 22, 19, 23, 20, 21]]) Note that when the ``out`` parameter is given, the return value is ``out``: >>> y is x True permutation(x, axis=0) Randomly permute a sequence, or return a permuted range. Parameters ---------- x : int or array_like If `x` is an integer, randomly permute ``np.arange(x)``. If `x` is an array, make a copy and shuffle the elements randomly. axis : int, optional The axis which `x` is shuffled along. Default is 0. Returns ------- out : ndarray Permuted sequence or array range. Examples -------- >>> rng = np.random.default_rng() >>> rng.permutation(10) array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6]) # random >>> rng.permutation([1, 4, 9, 12, 15]) array([15, 1, 9, 4, 12]) # random >>> arr = np.arange(9).reshape((3, 3)) >>> rng.permutation(arr) array([[6, 7, 8], # random [0, 1, 2], [3, 4, 5]]) >>> rng.permutation("abc") Traceback (most recent call last): ... numpy.exceptions.AxisError: axis 0 is out of bounds for array of dimension 0 >>> arr = np.arange(9).reshape((3, 3)) >>> rng.permutation(arr, axis=1) array([[0, 2, 1], # random [3, 5, 4], [6, 8, 7]]) pareto(a, size=None) Draw samples from a Pareto II (AKA Lomax) distribution with specified shape. Parameters ---------- a : float or array_like of floats Shape of the distribution. Must be positive. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` is a scalar. Otherwise, ``np.array(a).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the Pareto II distribution. See Also -------- scipy.stats.pareto : Pareto I distribution scipy.stats.lomax : Lomax (Pareto II) distribution scipy.stats.genpareto : Generalized Pareto distribution Notes ----- The probability density for the Pareto II distribution is .. math:: p(x) = \frac{a}{{x+1}^{a+1}} , x \ge 0 where :math:`a > 0` is the shape. The Pareto II distribution is a shifted and scaled version of the Pareto I distribution, which can be found in `scipy.stats.pareto`. References ---------- .. [1] Francis Hunt and Paul Johnson, On the Pareto Distribution of Sourceforge projects. .. [2] Pareto, V. (1896). Course of Political Economy. Lausanne. .. [3] Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme Values, Birkhauser Verlag, Basel, pp 23-30. .. [4] Wikipedia, "Pareto distribution", https://en.wikipedia.org/wiki/Pareto_distribution Examples -------- Draw samples from the distribution: >>> a = 3. >>> rng = np.random.default_rng() >>> s = rng.pareto(a, 10000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> x = np.linspace(0, 3, 50) >>> pdf = a / (x+1)**(a+1) >>> plt.hist(s, bins=x, density=True, label='histogram') >>> plt.plot(x, pdf, linewidth=2, color='r', label='pdf') >>> plt.xlim(x.min(), x.max()) >>> plt.legend() >>> plt.show() out must have the same shape as x normal(loc=0.0, scale=1.0, size=None) Draw random samples from a normal (Gaussian) distribution. The probability density function of the normal distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently [2]_, is often called the bell curve because of its characteristic shape (see the example below). The normal distributions occurs often in nature. For example, it describes the commonly occurring distribution of samples influenced by a large number of tiny, random disturbances, each with its own unique distribution [2]_. Parameters ---------- loc : float or array_like of floats Mean ("centre") of the distribution. scale : float or array_like of floats Standard deviation (spread or "width") of the distribution. Must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized normal distribution. See Also -------- scipy.stats.norm : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Gaussian distribution is .. math:: p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }} e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} }, where :math:`\mu` is the mean and :math:`\sigma` the standard deviation. The square of the standard deviation, :math:`\sigma^2`, is called the variance. The function has its peak at the mean, and its "spread" increases with the standard deviation (the function reaches 0.607 times its maximum at :math:`x + \sigma` and :math:`x - \sigma` [2]_). This implies that :meth:`normal` is more likely to return samples lying close to the mean, rather than those far away. References ---------- .. [1] Wikipedia, "Normal distribution", https://en.wikipedia.org/wiki/Normal_distribution .. [2] P. R. Peebles Jr., "Central Limit Theorem" in "Probability, Random Variables and Random Signal Principles", 4th ed., 2001, pp. 51, 51, 125. Examples -------- Draw samples from the distribution: >>> mu, sigma = 0, 0.1 # mean and standard deviation >>> rng = np.random.default_rng() >>> s = rng.normal(mu, sigma, 1000) Verify the mean and the variance: >>> abs(mu - np.mean(s)) 0.0 # may vary >>> abs(sigma - np.std(s, ddof=1)) 0.0 # may vary Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, 30, density=True) >>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) * ... np.exp( - (bins - mu)**2 / (2 * sigma**2) ), ... linewidth=2, color='r') >>> plt.show() Two-by-four array of samples from the normal distribution with mean 3 and standard deviation 2.5: >>> rng = np.random.default_rng() >>> rng.normal(3, 2.5, size=(2, 4)) array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random noncentral_f(dfnum, dfden, nonc, size=None) Draw samples from the noncentral F distribution. Samples are drawn from an F distribution with specified parameters, `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of freedom in denominator), where both parameters > 1. `nonc` is the non-centrality parameter. Parameters ---------- dfnum : float or array_like of floats Numerator degrees of freedom, must be > 0. .. versionchanged:: 1.14.0 Earlier NumPy versions required dfnum > 1. dfden : float or array_like of floats Denominator degrees of freedom, must be > 0. nonc : float or array_like of floats Non-centrality parameter, the sum of the squares of the numerator means, must be >= 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``dfnum``, ``dfden``, and ``nonc`` are all scalars. Otherwise, ``np.broadcast(dfnum, dfden, nonc).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized noncentral Fisher distribution. Notes ----- When calculating the power of an experiment (power = probability of rejecting the null hypothesis when a specific alternative is true) the non-central F statistic becomes important. When the null hypothesis is true, the F statistic follows a central F distribution. When the null hypothesis is not true, then it follows a non-central F statistic. References ---------- .. [1] Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NoncentralF-Distribution.html .. [2] Wikipedia, "Noncentral F-distribution", https://en.wikipedia.org/wiki/Noncentral_F-distribution Examples -------- In a study, testing for a specific alternative to the null hypothesis requires use of the Noncentral F distribution. We need to calculate the area in the tail of the distribution that exceeds the value of the F distribution for the null hypothesis. We'll plot the two probability distributions for comparison. >>> rng = np.random.default_rng() >>> dfnum = 3 # between group deg of freedom >>> dfden = 20 # within groups degrees of freedom >>> nonc = 3.0 >>> nc_vals = rng.noncentral_f(dfnum, dfden, nonc, 1000000) >>> NF = np.histogram(nc_vals, bins=50, density=True) >>> c_vals = rng.f(dfnum, dfden, 1000000) >>> F = np.histogram(c_vals, bins=50, density=True) >>> import matplotlib.pyplot as plt >>> plt.plot(F[1][1:], F[0]) >>> plt.plot(NF[1][1:], NF[0]) >>> plt.show() noncentral_chisquare(df, nonc, size=None) Draw samples from a noncentral chi-square distribution. The noncentral :math:`\chi^2` distribution is a generalization of the :math:`\chi^2` distribution. Parameters ---------- df : float or array_like of floats Degrees of freedom, must be > 0. .. versionchanged:: 1.10.0 Earlier NumPy versions required dfnum > 1. nonc : float or array_like of floats Non-centrality, must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``df`` and ``nonc`` are both scalars. Otherwise, ``np.broadcast(df, nonc).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized noncentral chi-square distribution. Notes ----- The probability density function for the noncentral Chi-square distribution is .. math:: P(x;df,nonc) = \sum^{\infty}_{i=0} \frac{e^{-nonc/2}(nonc/2)^{i}}{i!} P_{Y_{df+2i}}(x), where :math:`Y_{q}` is the Chi-square with q degrees of freedom. References ---------- .. [1] Wikipedia, "Noncentral chi-squared distribution" https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution Examples -------- Draw values from the distribution and plot the histogram >>> rng = np.random.default_rng() >>> import matplotlib.pyplot as plt >>> values = plt.hist(rng.noncentral_chisquare(3, 20, 100000), ... bins=200, density=True) >>> plt.show() Draw values from a noncentral chisquare with very small noncentrality, and compare to a chisquare. >>> plt.figure() >>> values = plt.hist(rng.noncentral_chisquare(3, .0000001, 100000), ... bins=np.arange(0., 25, .1), density=True) >>> values2 = plt.hist(rng.chisquare(3, 100000), ... bins=np.arange(0., 25, .1), density=True) >>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob') >>> plt.show() Demonstrate how large values of non-centrality lead to a more symmetric distribution. >>> plt.figure() >>> values = plt.hist(rng.noncentral_chisquare(3, 20, 100000), ... bins=200, density=True) >>> plt.show() negative_binomial(n, p, size=None) Draw samples from a negative binomial distribution. Samples are drawn from a negative binomial distribution with specified parameters, `n` successes and `p` probability of success where `n` is > 0 and `p` is in the interval (0, 1]. Parameters ---------- n : float or array_like of floats Parameter of the distribution, > 0. p : float or array_like of floats Parameter of the distribution. Must satisfy 0 < p <= 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``n`` and ``p`` are both scalars. Otherwise, ``np.broadcast(n, p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized negative binomial distribution, where each sample is equal to N, the number of failures that occurred before a total of n successes was reached. Notes ----- The probability mass function of the negative binomial distribution is .. math:: P(N;n,p) = \frac{\Gamma(N+n)}{N!\Gamma(n)}p^{n}(1-p)^{N}, where :math:`n` is the number of successes, :math:`p` is the probability of success, :math:`N+n` is the number of trials, and :math:`\Gamma` is the gamma function. When :math:`n` is an integer, :math:`\frac{\Gamma(N+n)}{N!\Gamma(n)} = \binom{N+n-1}{N}`, which is the more common form of this term in the pmf. The negative binomial distribution gives the probability of N failures given n successes, with a success on the last trial. If one throws a die repeatedly until the third time a "1" appears, then the probability distribution of the number of non-"1"s that appear before the third "1" is a negative binomial distribution. Because this method internally calls ``Generator.poisson`` with an intermediate random value, a ValueError is raised when the choice of :math:`n` and :math:`p` would result in the mean + 10 sigma of the sampled intermediate distribution exceeding the max acceptable value of the ``Generator.poisson`` method. This happens when :math:`p` is too low (a lot of failures happen for every success) and :math:`n` is too big ( a lot of successes are allowed). Therefore, the :math:`n` and :math:`p` values must satisfy the constraint: .. math:: n\frac{1-p}{p}+10n\sqrt{n}\frac{1-p}{p}<2^{63}-1-10\sqrt{2^{63}-1}, Where the left side of the equation is the derived mean + 10 sigma of a sample from the gamma distribution internally used as the :math:`lam` parameter of a poisson sample, and the right side of the equation is the constraint for maximum value of :math:`lam` in ``Generator.poisson``. References ---------- .. [1] Weisstein, Eric W. "Negative Binomial Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NegativeBinomialDistribution.html .. [2] Wikipedia, "Negative binomial distribution", https://en.wikipedia.org/wiki/Negative_binomial_distribution Examples -------- Draw samples from the distribution: A real world example. A company drills wild-cat oil exploration wells, each with an estimated probability of success of 0.1. What is the probability of having one success for each successive well, that is what is the probability of a single success after drilling 5 wells, after 6 wells, etc.? >>> rng = np.random.default_rng() >>> s = rng.negative_binomial(1, 0.1, 100000) >>> for i in range(1, 11): # doctest: +SKIP ... probability = sum(s>> mean = [0, 0] >>> cov = [[1, 0], [0, 100]] # diagonal covariance Diagonal covariance means that points are oriented along x or y-axis: >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() >>> x, y = rng.multivariate_normal(mean, cov, 5000).T >>> plt.plot(x, y, 'x') >>> plt.axis('equal') >>> plt.show() Note that the covariance matrix must be positive semidefinite (a.k.a. nonnegative-definite). Otherwise, the behavior of this method is undefined and backwards compatibility is not guaranteed. This function internally uses linear algebra routines, and thus results may not be identical (even up to precision) across architectures, OSes, or even builds. For example, this is likely if ``cov`` has multiple equal singular values and ``method`` is ``'svd'`` (default). In this case, ``method='cholesky'`` may be more robust. References ---------- .. [1] Papoulis, A., "Probability, Random Variables, and Stochastic Processes," 3rd ed., New York: McGraw-Hill, 1991. .. [2] Duda, R. O., Hart, P. E., and Stork, D. G., "Pattern Classification," 2nd ed., New York: Wiley, 2001. Examples -------- >>> mean = (1, 2) >>> cov = [[1, 0], [0, 1]] >>> rng = np.random.default_rng() >>> x = rng.multivariate_normal(mean, cov, (3, 3)) >>> x.shape (3, 3, 2) We can use a different method other than the default to factorize cov: >>> y = rng.multivariate_normal(mean, cov, (3, 3), method='cholesky') >>> y.shape (3, 3, 2) Here we generate 800 samples from the bivariate normal distribution with mean [0, 0] and covariance matrix [[6, -3], [-3, 3.5]]. The expected variances of the first and second components of the sample are 6 and 3.5, respectively, and the expected correlation coefficient is -3/sqrt(6*3.5) ≈ -0.65465. >>> cov = np.array([[6, -3], [-3, 3.5]]) >>> pts = rng.multivariate_normal([0, 0], cov, size=800) Check that the mean, covariance, and correlation coefficient of the sample are close to the expected values: >>> pts.mean(axis=0) array([ 0.0326911 , -0.01280782]) # may vary >>> np.cov(pts.T) array([[ 5.96202397, -2.85602287], [-2.85602287, 3.47613949]]) # may vary >>> np.corrcoef(pts.T)[0, 1] -0.6273591314603949 # may vary We can visualize this data with a scatter plot. The orientation of the point cloud illustrates the negative correlation of the components of this sample. >>> import matplotlib.pyplot as plt >>> plt.plot(pts[:, 0], pts[:, 1], '.', alpha=0.5) >>> plt.axis('equal') >>> plt.grid() >>> plt.show() multivariate_hypergeometric(colors, nsample, size=None, method='marginals') Generate variates from a multivariate hypergeometric distribution. The multivariate hypergeometric distribution is a generalization of the hypergeometric distribution. Choose ``nsample`` items at random without replacement from a collection with ``N`` distinct types. ``N`` is the length of ``colors``, and the values in ``colors`` are the number of occurrences of that type in the collection. The total number of items in the collection is ``sum(colors)``. Each random variate generated by this function is a vector of length ``N`` holding the counts of the different types that occurred in the ``nsample`` items. The name ``colors`` comes from a common description of the distribution: it is the probability distribution of the number of marbles of each color selected without replacement from an urn containing marbles of different colors; ``colors[i]`` is the number of marbles in the urn with color ``i``. Parameters ---------- colors : sequence of integers The number of each type of item in the collection from which a sample is drawn. The values in ``colors`` must be nonnegative. To avoid loss of precision in the algorithm, ``sum(colors)`` must be less than ``10**9`` when `method` is "marginals". nsample : int The number of items selected. ``nsample`` must not be greater than ``sum(colors)``. size : int or tuple of ints, optional The number of variates to generate, either an integer or a tuple holding the shape of the array of variates. If the given size is, e.g., ``(k, m)``, then ``k * m`` variates are drawn, where one variate is a vector of length ``len(colors)``, and the return value has shape ``(k, m, len(colors))``. If `size` is an integer, the output has shape ``(size, len(colors))``. Default is None, in which case a single variate is returned as an array with shape ``(len(colors),)``. method : string, optional Specify the algorithm that is used to generate the variates. Must be 'count' or 'marginals' (the default). See the Notes for a description of the methods. Returns ------- variates : ndarray Array of variates drawn from the multivariate hypergeometric distribution. See Also -------- hypergeometric : Draw samples from the (univariate) hypergeometric distribution. Notes ----- The two methods do not return the same sequence of variates. The "count" algorithm is roughly equivalent to the following numpy code:: choices = np.repeat(np.arange(len(colors)), colors) selection = np.random.choice(choices, nsample, replace=False) variate = np.bincount(selection, minlength=len(colors)) The "count" algorithm uses a temporary array of integers with length ``sum(colors)``. The "marginals" algorithm generates a variate by using repeated calls to the univariate hypergeometric sampler. It is roughly equivalent to:: variate = np.zeros(len(colors), dtype=np.int64) # `remaining` is the cumulative sum of `colors` from the last # element to the first; e.g. if `colors` is [3, 1, 5], then # `remaining` is [9, 6, 5]. remaining = np.cumsum(colors[::-1])[::-1] for i in range(len(colors)-1): if nsample < 1: break variate[i] = hypergeometric(colors[i], remaining[i+1], nsample) nsample -= variate[i] variate[-1] = nsample The default method is "marginals". For some cases (e.g. when `colors` contains relatively small integers), the "count" method can be significantly faster than the "marginals" method. If performance of the algorithm is important, test the two methods with typical inputs to decide which works best. .. versionadded:: 1.18.0 Examples -------- >>> colors = [16, 8, 4] >>> seed = 4861946401452 >>> gen = np.random.Generator(np.random.PCG64(seed)) >>> gen.multivariate_hypergeometric(colors, 6) array([5, 0, 1]) >>> gen.multivariate_hypergeometric(colors, 6, size=3) array([[5, 0, 1], [2, 2, 2], [3, 3, 0]]) >>> gen.multivariate_hypergeometric(colors, 6, size=(2, 2)) array([[[3, 2, 1], [3, 2, 1]], [[4, 1, 1], [3, 2, 1]]]) multinomial(n, pvals, size=None) Draw samples from a multinomial distribution. The multinomial distribution is a multivariate generalization of the binomial distribution. Take an experiment with one of ``p`` possible outcomes. An example of such an experiment is throwing a dice, where the outcome can be 1 through 6. Each sample drawn from the distribution represents `n` such experiments. Its values, ``X_i = [X_0, X_1, ..., X_p]``, represent the number of times the outcome was ``i``. Parameters ---------- n : int or array-like of ints Number of experiments. pvals : array-like of floats Probabilities of each of the ``p`` different outcomes with shape ``(k0, k1, ..., kn, p)``. Each element ``pvals[i,j,...,:]`` must sum to 1 (however, the last element is always assumed to account for the remaining probability, as long as ``sum(pvals[..., :-1], axis=-1) <= 1.0``. Must have at least 1 dimension where pvals.shape[-1] > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn each with ``p`` elements. Default is None where the output size is determined by the broadcast shape of ``n`` and all by the final dimension of ``pvals``, which is denoted as ``b=(b0, b1, ..., bq)``. If size is not None, then it must be compatible with the broadcast shape ``b``. Specifically, size must have ``q`` or more elements and size[-(q-j):] must equal ``bj``. Returns ------- out : ndarray The drawn samples, of shape size, if provided. When size is provided, the output shape is size + (p,) If not specified, the shape is determined by the broadcast shape of ``n`` and ``pvals``, ``(b0, b1, ..., bq)`` augmented with the dimension of the multinomial, ``p``, so that that output shape is ``(b0, b1, ..., bq, p)``. Each entry ``out[i,j,...,:]`` is a ``p``-dimensional value drawn from the distribution. .. versionchanged:: 1.22.0 Added support for broadcasting `pvals` against `n` Examples -------- Throw a dice 20 times: >>> rng = np.random.default_rng() >>> rng.multinomial(20, [1/6.]*6, size=1) array([[4, 1, 7, 5, 2, 1]]) # random It landed 4 times on 1, once on 2, etc. Now, throw the dice 20 times, and 20 times again: >>> rng.multinomial(20, [1/6.]*6, size=2) array([[3, 4, 3, 3, 4, 3], [2, 4, 3, 4, 0, 7]]) # random For the first run, we threw 3 times 1, 4 times 2, etc. For the second, we threw 2 times 1, 4 times 2, etc. Now, do one experiment throwing the dice 10 time, and 10 times again, and another throwing the dice 20 times, and 20 times again: >>> rng.multinomial([[10], [20]], [1/6.]*6, size=(2, 2)) array([[[2, 4, 0, 1, 2, 1], [1, 3, 0, 3, 1, 2]], [[1, 4, 4, 4, 4, 3], [3, 3, 2, 5, 5, 2]]]) # random The first array shows the outcomes of throwing the dice 10 times, and the second shows the outcomes from throwing the dice 20 times. A loaded die is more likely to land on number 6: >>> rng.multinomial(100, [1/7.]*5 + [2/7.]) array([11, 16, 14, 17, 16, 26]) # random Simulate 10 throws of a 4-sided die and 20 throws of a 6-sided die >>> rng.multinomial([10, 20],[[1/4]*4 + [0]*2, [1/6]*6]) array([[2, 1, 4, 3, 0, 0], [3, 3, 3, 6, 1, 4]], dtype=int64) # random Generate categorical random variates from two categories where the first has 3 outcomes and the second has 2. >>> rng.multinomial(1, [[.1, .5, .4 ], [.3, .7, .0]]) array([[0, 0, 1], [0, 1, 0]], dtype=int64) # random ``argmax(axis=-1)`` is then used to return the categories. >>> pvals = [[.1, .5, .4 ], [.3, .7, .0]] >>> rvs = rng.multinomial(1, pvals, size=(4,2)) >>> rvs.argmax(axis=-1) array([[0, 1], [2, 0], [2, 1], [2, 0]], dtype=int64) # random The same output dimension can be produced using broadcasting. >>> rvs = rng.multinomial([[1]] * 4, pvals) >>> rvs.argmax(axis=-1) array([[0, 1], [2, 0], [2, 1], [2, 0]], dtype=int64) # random The probability inputs should be normalized. As an implementation detail, the value of the last entry is ignored and assumed to take up any leftover probability mass, but this should not be relied on. A biased coin which has twice as much weight on one side as on the other should be sampled like so: >>> rng.multinomial(100, [1.0 / 3, 2.0 / 3]) # RIGHT array([38, 62]) # random not like: >>> rng.multinomial(100, [1.0, 2.0]) # WRONG Traceback (most recent call last): ValueError: pvals < 0, pvals > 1 or pvals contains NaNs logseries(p, size=None) Draw samples from a logarithmic series distribution. Samples are drawn from a log series distribution with specified shape parameter, 0 <= ``p`` < 1. Parameters ---------- p : float or array_like of floats Shape parameter for the distribution. Must be in the range [0, 1). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``p`` is a scalar. Otherwise, ``np.array(p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized logarithmic series distribution. See Also -------- scipy.stats.logser : probability density function, distribution or cumulative density function, etc. Notes ----- The probability mass function for the Log Series distribution is .. math:: P(k) = \frac{-p^k}{k \ln(1-p)}, where p = probability. The log series distribution is frequently used to represent species richness and occurrence, first proposed by Fisher, Corbet, and Williams in 1943 [2]. It may also be used to model the numbers of occupants seen in cars [3]. References ---------- .. [1] Buzas, Martin A.; Culver, Stephen J., Understanding regional species diversity through the log series distribution of occurrences: BIODIVERSITY RESEARCH Diversity & Distributions, Volume 5, Number 5, September 1999 , pp. 187-195(9). .. [2] Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology, 12:42-58. .. [3] D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small Data Sets, CRC Press, 1994. .. [4] Wikipedia, "Logarithmic distribution", https://en.wikipedia.org/wiki/Logarithmic_distribution Examples -------- Draw samples from the distribution: >>> a = .6 >>> rng = np.random.default_rng() >>> s = rng.logseries(a, 10000) >>> import matplotlib.pyplot as plt >>> bins = np.arange(-.5, max(s) + .5 ) >>> count, bins, _ = plt.hist(s, bins=bins, label='Sample count') # plot against distribution >>> def logseries(k, p): ... return -p**k/(k*np.log(1-p)) >>> centres = np.arange(1, max(s) + 1) >>> plt.plot(centres, logseries(centres, a) * s.size, 'r', label='logseries PMF') >>> plt.legend() >>> plt.show() lognormal(mean=0.0, sigma=1.0, size=None) Draw samples from a log-normal distribution. Draw samples from a log-normal distribution with specified mean, standard deviation, and array shape. Note that the mean and standard deviation are not the values for the distribution itself, but of the underlying normal distribution it is derived from. Parameters ---------- mean : float or array_like of floats, optional Mean value of the underlying normal distribution. Default is 0. sigma : float or array_like of floats, optional Standard deviation of the underlying normal distribution. Must be non-negative. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``mean`` and ``sigma`` are both scalars. Otherwise, ``np.broadcast(mean, sigma).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized log-normal distribution. See Also -------- scipy.stats.lognorm : probability density function, distribution, cumulative density function, etc. Notes ----- A variable `x` has a log-normal distribution if `log(x)` is normally distributed. The probability density function for the log-normal distribution is: .. math:: p(x) = \frac{1}{\sigma x \sqrt{2\pi}} e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})} where :math:`\mu` is the mean and :math:`\sigma` is the standard deviation of the normally distributed logarithm of the variable. A log-normal distribution results if a random variable is the *product* of a large number of independent, identically-distributed variables in the same way that a normal distribution results if the variable is the *sum* of a large number of independent, identically-distributed variables. References ---------- .. [1] Limpert, E., Stahel, W. A., and Abbt, M., "Log-normal Distributions across the Sciences: Keys and Clues," BioScience, Vol. 51, No. 5, May, 2001. https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf .. [2] Reiss, R.D. and Thomas, M., "Statistical Analysis of Extreme Values," Basel: Birkhauser Verlag, 2001, pp. 31-32. Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> mu, sigma = 3., 1. # mean and standard deviation >>> s = rng.lognormal(mu, sigma, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, 100, density=True, align='mid') >>> x = np.linspace(min(bins), max(bins), 10000) >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2)) ... / (x * sigma * np.sqrt(2 * np.pi))) >>> plt.plot(x, pdf, linewidth=2, color='r') >>> plt.axis('tight') >>> plt.show() Demonstrate that taking the products of random samples from a uniform distribution can be fit well by a log-normal probability density function. >>> # Generate a thousand samples: each is the product of 100 random >>> # values, drawn from a normal distribution. >>> rng = rng >>> b = [] >>> for i in range(1000): ... a = 10. + rng.standard_normal(100) ... b.append(np.prod(a)) >>> b = np.array(b) / np.min(b) # scale values to be positive >>> count, bins, _ = plt.hist(b, 100, density=True, align='mid') >>> sigma = np.std(np.log(b)) >>> mu = np.mean(np.log(b)) >>> x = np.linspace(min(bins), max(bins), 10000) >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2)) ... / (x * sigma * np.sqrt(2 * np.pi))) >>> plt.plot(x, pdf, color='r', linewidth=2) >>> plt.show() logistic(loc=0.0, scale=1.0, size=None) Draw samples from a logistic distribution. Samples are drawn from a logistic distribution with specified parameters, loc (location or mean, also median), and scale (>0). Parameters ---------- loc : float or array_like of floats, optional Parameter of the distribution. Default is 0. scale : float or array_like of floats, optional Parameter of the distribution. Must be non-negative. Default is 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized logistic distribution. See Also -------- scipy.stats.logistic : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Logistic distribution is .. math:: P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2}, where :math:`\mu` = location and :math:`s` = scale. The Logistic distribution is used in Extreme Value problems where it can act as a mixture of Gumbel distributions, in Epidemiology, and by the World Chess Federation (FIDE) where it is used in the Elo ranking system, assuming the performance of each player is a logistically distributed random variable. References ---------- .. [1] Reiss, R.-D. and Thomas M. (2001), "Statistical Analysis of Extreme Values, from Insurance, Finance, Hydrology and Other Fields," Birkhauser Verlag, Basel, pp 132-133. .. [2] Weisstein, Eric W. "Logistic Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LogisticDistribution.html .. [3] Wikipedia, "Logistic-distribution", https://en.wikipedia.org/wiki/Logistic_distribution Examples -------- Draw samples from the distribution: >>> loc, scale = 10, 1 >>> rng = np.random.default_rng() >>> s = rng.logistic(loc, scale, 10000) >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, bins=50, label='Sampled data') # plot sampled data against the exact distribution >>> def logistic(x, loc, scale): ... return np.exp((loc-x)/scale)/(scale*(1+np.exp((loc-x)/scale))**2) >>> logistic_values = logistic(bins, loc, scale) >>> bin_spacing = np.mean(np.diff(bins)) >>> plt.plot(bins, logistic_values * bin_spacing * s.size, label='Logistic PDF') >>> plt.legend() >>> plt.show() is not compatible with broadcast dimensions of inputs integers(low, high=None, size=None, dtype=np.int64, endpoint=False) Return random integers from `low` (inclusive) to `high` (exclusive), or if endpoint=True, `low` (inclusive) to `high` (inclusive). Replaces `RandomState.randint` (with endpoint=False) and `RandomState.random_integers` (with endpoint=True) Return random integers from the "discrete uniform" distribution of the specified dtype. If `high` is None (the default), then results are from 0 to `low`. Parameters ---------- low : int or array-like of ints Lowest (signed) integers to be drawn from the distribution (unless ``high=None``, in which case this parameter is 0 and this value is used for `high`). high : int or array-like of ints, optional If provided, one above the largest (signed) integer to be drawn from the distribution (see above for behavior if ``high=None``). If array-like, must contain integer values size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. dtype : dtype, optional Desired dtype of the result. Byteorder must be native. The default value is np.int64. endpoint : bool, optional If true, sample from the interval [low, high] instead of the default [low, high) Defaults to False Returns ------- out : int or ndarray of ints `size`-shaped array of random integers from the appropriate distribution, or a single such random int if `size` not provided. Notes ----- When using broadcasting with uint64 dtypes, the maximum value (2**64) cannot be represented as a standard integer type. The high array (or low if high is None) must have object dtype, e.g., array([2**64]). Examples -------- >>> rng = np.random.default_rng() >>> rng.integers(2, size=10) array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) # random >>> rng.integers(1, size=10) array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0]) Generate a 2 x 4 array of ints between 0 and 4, inclusive: >>> rng.integers(5, size=(2, 4)) array([[4, 0, 2, 1], [3, 2, 2, 0]]) # random Generate a 1 x 3 array with 3 different upper bounds >>> rng.integers(1, [3, 5, 10]) array([2, 2, 9]) # random Generate a 1 by 3 array with 3 different lower bounds >>> rng.integers([1, 5, 7], 10) array([9, 8, 7]) # random Generate a 2 by 4 array using broadcasting with dtype of uint8 >>> rng.integers([1, 3, 5, 7], [[10], [20]], dtype=np.uint8) array([[ 8, 6, 9, 7], [ 1, 16, 9, 12]], dtype=uint8) # random References ---------- .. [1] Daniel Lemire., "Fast Random Integer Generation in an Interval", ACM Transactions on Modeling and Computer Simulation 29 (1), 2019, https://arxiv.org/abs/1805.10941. hypergeometric(ngood, nbad, nsample, size=None) Draw samples from a Hypergeometric distribution. Samples are drawn from a hypergeometric distribution with specified parameters, `ngood` (ways to make a good selection), `nbad` (ways to make a bad selection), and `nsample` (number of items sampled, which is less than or equal to the sum ``ngood + nbad``). Parameters ---------- ngood : int or array_like of ints Number of ways to make a good selection. Must be nonnegative and less than 10**9. nbad : int or array_like of ints Number of ways to make a bad selection. Must be nonnegative and less than 10**9. nsample : int or array_like of ints Number of items sampled. Must be nonnegative and less than ``ngood + nbad``. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if `ngood`, `nbad`, and `nsample` are all scalars. Otherwise, ``np.broadcast(ngood, nbad, nsample).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized hypergeometric distribution. Each sample is the number of good items within a randomly selected subset of size `nsample` taken from a set of `ngood` good items and `nbad` bad items. See Also -------- multivariate_hypergeometric : Draw samples from the multivariate hypergeometric distribution. scipy.stats.hypergeom : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Hypergeometric distribution is .. math:: P(x) = \frac{\binom{g}{x}\binom{b}{n-x}}{\binom{g+b}{n}}, where :math:`0 \le x \le n` and :math:`n-b \le x \le g` for P(x) the probability of ``x`` good results in the drawn sample, g = `ngood`, b = `nbad`, and n = `nsample`. Consider an urn with black and white marbles in it, `ngood` of them are black and `nbad` are white. If you draw `nsample` balls without replacement, then the hypergeometric distribution describes the distribution of black balls in the drawn sample. Note that this distribution is very similar to the binomial distribution, except that in this case, samples are drawn without replacement, whereas in the Binomial case samples are drawn with replacement (or the sample space is infinite). As the sample space becomes large, this distribution approaches the binomial. The arguments `ngood` and `nbad` each must be less than `10**9`. For extremely large arguments, the algorithm that is used to compute the samples [4]_ breaks down because of loss of precision in floating point calculations. For such large values, if `nsample` is not also large, the distribution can be approximated with the binomial distribution, `binomial(n=nsample, p=ngood/(ngood + nbad))`. References ---------- .. [1] Lentner, Marvin, "Elementary Applied Statistics", Bogden and Quigley, 1972. .. [2] Weisstein, Eric W. "Hypergeometric Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HypergeometricDistribution.html .. [3] Wikipedia, "Hypergeometric distribution", https://en.wikipedia.org/wiki/Hypergeometric_distribution .. [4] Stadlober, Ernst, "The ratio of uniforms approach for generating discrete random variates", Journal of Computational and Applied Mathematics, 31, pp. 181-189 (1990). Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> ngood, nbad, nsamp = 100, 2, 10 # number of good, number of bad, and number of samples >>> s = rng.hypergeometric(ngood, nbad, nsamp, 1000) >>> from matplotlib.pyplot import hist >>> hist(s) # note that it is very unlikely to grab both bad items Suppose you have an urn with 15 white and 15 black marbles. If you pull 15 marbles at random, how likely is it that 12 or more of them are one color? >>> s = rng.hypergeometric(15, 15, 15, 100000) >>> sum(s>=12)/100000. + sum(s<=3)/100000. # answer = 0.003 ... pretty unlikely! gumbel(loc=0.0, scale=1.0, size=None) Draw samples from a Gumbel distribution. Draw samples from a Gumbel distribution with specified location and scale. For more information on the Gumbel distribution, see Notes and References below. Parameters ---------- loc : float or array_like of floats, optional The location of the mode of the distribution. Default is 0. scale : float or array_like of floats, optional The scale parameter of the distribution. Default is 1. Must be non- negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Gumbel distribution. See Also -------- scipy.stats.gumbel_l scipy.stats.gumbel_r scipy.stats.genextreme weibull Notes ----- The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme Value Type I) distribution is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. The Gumbel is a special case of the Extreme Value Type I distribution for maximums from distributions with "exponential-like" tails. The probability density for the Gumbel distribution is .. math:: p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/ \beta}}, where :math:`\mu` is the mode, a location parameter, and :math:`\beta` is the scale parameter. The Gumbel (named for German mathematician Emil Julius Gumbel) was used very early in the hydrology literature, for modeling the occurrence of flood events. It is also used for modeling maximum wind speed and rainfall rates. It is a "fat-tailed" distribution - the probability of an event in the tail of the distribution is larger than if one used a Gaussian, hence the surprisingly frequent occurrence of 100-year floods. Floods were initially modeled as a Gaussian process, which underestimated the frequency of extreme events. It is one of a class of extreme value distributions, the Generalized Extreme Value (GEV) distributions, which also includes the Weibull and Frechet. The function has a mean of :math:`\mu + 0.57721\beta` and a variance of :math:`\frac{\pi^2}{6}\beta^2`. References ---------- .. [1] Gumbel, E. J., "Statistics of Extremes," New York: Columbia University Press, 1958. .. [2] Reiss, R.-D. and Thomas, M., "Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields," Basel: Birkhauser Verlag, 2001. Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> mu, beta = 0, 0.1 # location and scale >>> s = rng.gumbel(mu, beta, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, 30, density=True) >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta) ... * np.exp( -np.exp( -(bins - mu) /beta) ), ... linewidth=2, color='r') >>> plt.show() Show how an extreme value distribution can arise from a Gaussian process and compare to a Gaussian: >>> means = [] >>> maxima = [] >>> for i in range(0,1000) : ... a = rng.normal(mu, beta, 1000) ... means.append(a.mean()) ... maxima.append(a.max()) >>> count, bins, _ = plt.hist(maxima, 30, density=True) >>> beta = np.std(maxima) * np.sqrt(6) / np.pi >>> mu = np.mean(maxima) - 0.57721*beta >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta) ... * np.exp(-np.exp(-(bins - mu)/beta)), ... linewidth=2, color='r') >>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi)) ... * np.exp(-(bins - mu)**2 / (2 * beta**2)), ... linewidth=2, color='g') >>> plt.show() geometric(p, size=None) Draw samples from the geometric distribution. Bernoulli trials are experiments with one of two outcomes: success or failure (an example of such an experiment is flipping a coin). The geometric distribution models the number of trials that must be run in order to achieve success. It is therefore supported on the positive integers, ``k = 1, 2, ...``. The probability mass function of the geometric distribution is .. math:: f(k) = (1 - p)^{k - 1} p where `p` is the probability of success of an individual trial. Parameters ---------- p : float or array_like of floats The probability of success of an individual trial. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``p`` is a scalar. Otherwise, ``np.array(p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized geometric distribution. References ---------- .. [1] Wikipedia, "Geometric distribution", https://en.wikipedia.org/wiki/Geometric_distribution Examples -------- Draw 10,000 values from the geometric distribution, with the probability of an individual success equal to ``p = 0.35``: >>> p, size = 0.35, 10000 >>> rng = np.random.default_rng() >>> sample = rng.geometric(p=p, size=size) What proportion of trials succeeded after a single run? >>> (sample == 1).sum()/size 0.34889999999999999 # may vary The geometric distribution with ``p=0.35`` looks as follows: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(sample, bins=30, density=True) >>> plt.plot(bins, (1-p)**(bins-1)*p) >>> plt.xlim([0, 25]) >>> plt.show() gamma(shape, scale=1.0, size=None) Draw samples from a Gamma distribution. Samples are drawn from a Gamma distribution with specified parameters, `shape` (sometimes designated "k") and `scale` (sometimes designated "theta"), where both parameters are > 0. Parameters ---------- shape : float or array_like of floats The shape of the gamma distribution. Must be non-negative. scale : float or array_like of floats, optional The scale of the gamma distribution. Must be non-negative. Default is equal to 1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``shape`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(shape, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized gamma distribution. See Also -------- scipy.stats.gamma : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Gamma distribution is .. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)}, where :math:`k` is the shape and :math:`\theta` the scale, and :math:`\Gamma` is the Gamma function. The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant. References ---------- .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GammaDistribution.html .. [2] Wikipedia, "Gamma distribution", https://en.wikipedia.org/wiki/Gamma_distribution Examples -------- Draw samples from the distribution: >>> shape, scale = 2., 2. # mean=4, std=2*sqrt(2) >>> rng = np.random.default_rng() >>> s = rng.gamma(shape, scale, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> import scipy.special as sps # doctest: +SKIP >>> count, bins, _ = plt.hist(s, 50, density=True) >>> y = bins**(shape-1)*(np.exp(-bins/scale) / # doctest: +SKIP ... (sps.gamma(shape)*scale**shape)) >>> plt.plot(bins, y, linewidth=2, color='r') # doctest: +SKIP >>> plt.show() f(dfnum, dfden, size=None) Draw samples from an F distribution. Samples are drawn from an F distribution with specified parameters, `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of freedom in denominator), where both parameters must be greater than zero. The random variate of the F distribution (also known as the Fisher distribution) is a continuous probability distribution that arises in ANOVA tests, and is the ratio of two chi-square variates. Parameters ---------- dfnum : float or array_like of floats Degrees of freedom in numerator, must be > 0. dfden : float or array_like of float Degrees of freedom in denominator, must be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``dfnum`` and ``dfden`` are both scalars. Otherwise, ``np.broadcast(dfnum, dfden).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Fisher distribution. See Also -------- scipy.stats.f : probability density function, distribution or cumulative density function, etc. Notes ----- The F statistic is used to compare in-group variances to between-group variances. Calculating the distribution depends on the sampling, and so it is a function of the respective degrees of freedom in the problem. The variable `dfnum` is the number of samples minus one, the between-groups degrees of freedom, while `dfden` is the within-groups degrees of freedom, the sum of the number of samples in each group minus the number of groups. References ---------- .. [1] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill, Fifth Edition, 2002. .. [2] Wikipedia, "F-distribution", https://en.wikipedia.org/wiki/F-distribution Examples -------- An example from Glantz[1], pp 47-40: Two groups, children of diabetics (25 people) and children from people without diabetes (25 controls). Fasting blood glucose was measured, case group had a mean value of 86.1, controls had a mean value of 82.2. Standard deviations were 2.09 and 2.49 respectively. Are these data consistent with the null hypothesis that the parents diabetic status does not affect their children's blood glucose levels? Calculating the F statistic from the data gives a value of 36.01. Draw samples from the distribution: >>> dfnum = 1. # between group degrees of freedom >>> dfden = 48. # within groups degrees of freedom >>> rng = np.random.default_rng() >>> s = rng.f(dfnum, dfden, 1000) The lower bound for the top 1% of the samples is : >>> np.sort(s)[-10] 7.61988120985 # random So there is about a 1% chance that the F statistic will exceed 7.62, the measured value is 36, so the null hypothesis is rejected at the 1% level. The corresponding probability density function for ``n = 20`` and ``m = 20`` is: >>> import matplotlib.pyplot as plt >>> from scipy import stats # doctest: +SKIP >>> dfnum, dfden, size = 20, 20, 10000 >>> s = rng.f(dfnum=dfnum, dfden=dfden, size=size) >>> bins, density, _ = plt.hist(s, 30, density=True) >>> x = np.linspace(0, 5, 1000) >>> plt.plot(x, stats.f.pdf(x, dfnum, dfden)) >>> plt.xlim([0, 5]) >>> plt.show() exponential(scale=1.0, size=None) Draw samples from an exponential distribution. Its probability density function is .. math:: f(x; \frac{1}{\beta}) = \frac{1}{\beta} \exp(-\frac{x}{\beta}), for ``x > 0`` and 0 elsewhere. :math:`\beta` is the scale parameter, which is the inverse of the rate parameter :math:`\lambda = 1/\beta`. The rate parameter is an alternative, widely used parameterization of the exponential distribution [3]_. The exponential distribution is a continuous analogue of the geometric distribution. It describes many common situations, such as the size of raindrops measured over many rainstorms [1]_, or the time between page requests to Wikipedia [2]_. Parameters ---------- scale : float or array_like of floats The scale parameter, :math:`\beta = 1/\lambda`. Must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``scale`` is a scalar. Otherwise, ``np.array(scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized exponential distribution. Examples -------- Assume a company has 10000 customer support agents and the time between customer calls is exponentially distributed and that the average time between customer calls is 4 minutes. >>> scale, size = 4, 10000 >>> rng = np.random.default_rng() >>> time_between_calls = rng.exponential(scale=scale, size=size) What is the probability that a customer will call in the next 4 to 5 minutes? >>> x = ((time_between_calls < 5).sum())/size >>> y = ((time_between_calls < 4).sum())/size >>> x - y 0.08 # may vary The corresponding distribution can be visualized as follows: >>> import matplotlib.pyplot as plt >>> scale, size = 4, 10000 >>> rng = np.random.default_rng() >>> sample = rng.exponential(scale=scale, size=size) >>> count, bins, _ = plt.hist(sample, 30, density=True) >>> plt.plot(bins, scale**(-1)*np.exp(-scale**-1*bins), linewidth=2, color='r') >>> plt.show() References ---------- .. [1] Peyton Z. Peebles Jr., "Probability, Random Variables and Random Signal Principles", 4th ed, 2001, p. 57. .. [2] Wikipedia, "Poisson process", https://en.wikipedia.org/wiki/Poisson_process .. [3] Wikipedia, "Exponential distribution", https://en.wikipedia.org/wiki/Exponential_distribution choice(a, size=None, replace=True, p=None, axis=0, shuffle=True) Generates a random sample from a given array Parameters ---------- a : {array_like, int} If an ndarray, a random sample is generated from its elements. If an int, the random sample is generated from np.arange(a). size : {int, tuple[int]}, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn from the 1-d `a`. If `a` has more than one dimension, the `size` shape will be inserted into the `axis` dimension, so the output ``ndim`` will be ``a.ndim - 1 + len(size)``. Default is None, in which case a single value is returned. replace : bool, optional Whether the sample is with or without replacement. Default is True, meaning that a value of ``a`` can be selected multiple times. p : 1-D array_like, optional The probabilities associated with each entry in a. If not given, the sample assumes a uniform distribution over all entries in ``a``. axis : int, optional The axis along which the selection is performed. The default, 0, selects by row. shuffle : bool, optional Whether the sample is shuffled when sampling without replacement. Default is True, False provides a speedup. Returns ------- samples : single item or ndarray The generated random samples Raises ------ ValueError If a is an int and less than zero, if p is not 1-dimensional, if a is array-like with a size 0, if p is not a vector of probabilities, if a and p have different lengths, or if replace=False and the sample size is greater than the population size. See Also -------- integers, shuffle, permutation Notes ----- Setting user-specified probabilities through ``p`` uses a more general but less efficient sampler than the default. The general sampler produces a different sample than the optimized sampler even if each element of ``p`` is 1 / len(a). ``p`` must sum to 1 when cast to ``float64``. To ensure this, you may wish to normalize using ``p = p / np.sum(p, dtype=float)``. Examples -------- Generate a uniform random sample from np.arange(5) of size 3: >>> rng = np.random.default_rng() >>> rng.choice(5, 3) array([0, 3, 4]) # random >>> #This is equivalent to rng.integers(0,5,3) Generate a non-uniform random sample from np.arange(5) of size 3: >>> rng.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0]) array([3, 3, 0]) # random Generate a uniform random sample from np.arange(5) of size 3 without replacement: >>> rng.choice(5, 3, replace=False) array([3,1,0]) # random >>> #This is equivalent to rng.permutation(np.arange(5))[:3] Generate a uniform random sample from a 2-D array along the first axis (the default), without replacement: >>> rng.choice([[0, 1, 2], [3, 4, 5], [6, 7, 8]], 2, replace=False) array([[3, 4, 5], # random [0, 1, 2]]) Generate a non-uniform random sample from np.arange(5) of size 3 without replacement: >>> rng.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0]) array([2, 3, 0]) # random Any of the above can be repeated with an arbitrary array-like instead of just integers. For instance: >>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher'] >>> rng.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3]) array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], # random dtype=' 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``df`` is a scalar. Otherwise, ``np.array(df).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized chi-square distribution. Raises ------ ValueError When `df` <= 0 or when an inappropriate `size` (e.g. ``size=-1``) is given. Notes ----- The variable obtained by summing the squares of `df` independent, standard normally distributed random variables: .. math:: Q = \sum_{i=0}^{\mathtt{df}} X^2_i is chi-square distributed, denoted .. math:: Q \sim \chi^2_k. The probability density function of the chi-squared distribution is .. math:: p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}, where :math:`\Gamma` is the gamma function, .. math:: \Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt. References ---------- .. [1] NIST "Engineering Statistics Handbook" https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm Examples -------- >>> rng = np.random.default_rng() >>> rng.chisquare(2,4) array([ 1.89920014, 9.00867716, 3.13710533, 5.62318272]) # random The distribution of a chi-square random variable with 20 degrees of freedom looks as follows: >>> import matplotlib.pyplot as plt >>> import scipy.stats as stats >>> s = rng.chisquare(20, 10000) >>> count, bins, _ = plt.hist(s, 30, density=True) >>> x = np.linspace(0, 60, 1000) >>> plt.plot(x, stats.chi2.pdf(x, df=20)) >>> plt.xlim([0, 60]) >>> plt.show() bytes(length) Return random bytes. Parameters ---------- length : int Number of random bytes. Returns ------- out : bytes String of length `length`. Notes ----- This function generates random bytes from a discrete uniform distribution. The generated bytes are independent from the CPU's native endianness. Examples -------- >>> rng = np.random.default_rng() >>> rng.bytes(10) b'\xfeC\x9b\x86\x17\xf2\xa1\xafcp' # random binomial(n, p, size=None) Draw samples from a binomial distribution. Samples are drawn from a binomial distribution with specified parameters, n trials and p probability of success where n an integer >= 0 and p is in the interval [0,1]. (n may be input as a float, but it is truncated to an integer in use) Parameters ---------- n : int or array_like of ints Parameter of the distribution, >= 0. Floats are also accepted, but they will be truncated to integers. p : float or array_like of floats Parameter of the distribution, >= 0 and <=1. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``n`` and ``p`` are both scalars. Otherwise, ``np.broadcast(n, p).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized binomial distribution, where each sample is equal to the number of successes over the n trials. See Also -------- scipy.stats.binom : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the binomial distribution is .. math:: P(N) = \binom{n}{N}p^N(1-p)^{n-N}, where :math:`n` is the number of trials, :math:`p` is the probability of success, and :math:`N` is the number of successes. When estimating the standard error of a proportion in a population by using a random sample, the normal distribution works well unless the product p*n <=5, where p = population proportion estimate, and n = number of samples, in which case the binomial distribution is used instead. For example, a sample of 15 people shows 4 who are left handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4, so the binomial distribution should be used in this case. References ---------- .. [1] Dalgaard, Peter, "Introductory Statistics with R", Springer-Verlag, 2002. .. [2] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill, Fifth Edition, 2002. .. [3] Lentner, Marvin, "Elementary Applied Statistics", Bogden and Quigley, 1972. .. [4] Weisstein, Eric W. "Binomial Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BinomialDistribution.html .. [5] Wikipedia, "Binomial distribution", https://en.wikipedia.org/wiki/Binomial_distribution Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> n, p, size = 10, .5, 10000 >>> s = rng.binomial(n, p, 10000) Assume a company drills 9 wild-cat oil exploration wells, each with an estimated probability of success of ``p=0.1``. All nine wells fail. What is the probability of that happening? Over ``size = 20,000`` trials the probability of this happening is on average: >>> n, p, size = 9, 0.1, 20000 >>> np.sum(rng.binomial(n=n, p=p, size=size) == 0)/size 0.39015 # may vary The following can be used to visualize a sample with ``n=100``, ``p=0.4`` and the corresponding probability density function: >>> import matplotlib.pyplot as plt >>> from scipy.stats import binom >>> n, p, size = 100, 0.4, 10000 >>> sample = rng.binomial(n, p, size=size) >>> count, bins, _ = plt.hist(sample, 30, density=True) >>> x = np.arange(n) >>> y = binom.pmf(x, n, p) >>> plt.plot(x, y, linewidth=2, color='r') beta(a, b, size=None) Draw samples from a Beta distribution. The Beta distribution is a special case of the Dirichlet distribution, and is related to the Gamma distribution. It has the probability distribution function .. math:: f(x; a,b) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1} (1 - x)^{\beta - 1}, where the normalization, B, is the beta function, .. math:: B(\alpha, \beta) = \int_0^1 t^{\alpha - 1} (1 - t)^{\beta - 1} dt. It is often seen in Bayesian inference and order statistics. Parameters ---------- a : float or array_like of floats Alpha, positive (>0). b : float or array_like of floats Beta, positive (>0). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``a`` and ``b`` are both scalars. Otherwise, ``np.broadcast(a, b).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized beta distribution. Examples -------- The beta distribution has mean a/(a+b). If ``a == b`` and both are > 1, the distribution is symmetric with mean 0.5. >>> rng = np.random.default_rng() >>> a, b, size = 2.0, 2.0, 10000 >>> sample = rng.beta(a=a, b=b, size=size) >>> np.mean(sample) 0.5047328775385895 # may vary Otherwise the distribution is skewed left or right according to whether ``a`` or ``b`` is greater. The distribution is mirror symmetric. See for example: >>> a, b, size = 2, 7, 10000 >>> sample_left = rng.beta(a=a, b=b, size=size) >>> sample_right = rng.beta(a=b, b=a, size=size) >>> m_left, m_right = np.mean(sample_left), np.mean(sample_right) >>> print(m_left, m_right) 0.2238596793678923 0.7774613834041182 # may vary >>> print(m_left - a/(a+b)) 0.001637457145670096 # may vary >>> print(m_right - b/(a+b)) -0.0003163943736596009 # may vary Display the histogram of the two samples: >>> import matplotlib.pyplot as plt >>> plt.hist([sample_left, sample_right], ... 50, density=True, histtype='bar') >>> plt.show() References ---------- .. [1] Wikipedia, "Beta distribution", https://en.wikipedia.org/wiki/Beta_distribution .astype(np.float64)) > 1.0. The pvals array is cast to 64-bit floating point prior to checking the sum. Precision changes when casting may cause problems even if the sum of the original pvals is valid.When method is 'count', sum(colors) must not exceed %dGenerator.permutation (line 4927)Generator.multinomial (line 3963)Axis argument is only supported on ndarray objects wald(mean, scale, size=None) Draw samples from a Wald, or inverse Gaussian, distribution. As the scale approaches infinity, the distribution becomes more like a Gaussian. Some references claim that the Wald is an inverse Gaussian with mean equal to 1, but this is by no means universal. The inverse Gaussian distribution was first studied in relationship to Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian because there is an inverse relationship between the time to cover a unit distance and distance covered in unit time. Parameters ---------- mean : float or array_like of floats Distribution mean, must be > 0. scale : float or array_like of floats Scale parameter, must be > 0. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``mean`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(mean, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Wald distribution. Notes ----- The probability density function for the Wald distribution is .. math:: P(x;mean,scale) = \sqrt{\frac{scale}{2\pi x^3}}e^ \frac{-scale(x-mean)^2}{2\cdotp mean^2x} As noted above the inverse Gaussian distribution first arise from attempts to model Brownian motion. It is also a competitor to the Weibull for use in reliability modeling and modeling stock returns and interest rate processes. References ---------- .. [1] Brighton Webs Ltd., Wald Distribution, https://web.archive.org/web/20090423014010/http://www.brighton-webs.co.uk:80/distributions/wald.asp .. [2] Chhikara, Raj S., and Folks, J. Leroy, "The Inverse Gaussian Distribution: Theory : Methodology, and Applications", CRC Press, 1988. .. [3] Wikipedia, "Inverse Gaussian distribution" https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Examples -------- Draw values from the distribution and plot the histogram: >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() >>> h = plt.hist(rng.wald(3, 2, 100000), bins=200, density=True) >>> plt.show() standard_gamma(shape, size=None, dtype=np.float64, out=None) Draw samples from a standard Gamma distribution. Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated "k") and scale=1. Parameters ---------- shape : float or array_like of floats Parameter, must be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``shape`` is a scalar. Otherwise, ``np.array(shape).size`` samples are drawn. dtype : dtype, optional Desired dtype of the result, only `float64` and `float32` are supported. Byteorder must be native. The default value is np.float64. out : ndarray, optional Alternative output array in which to place the result. If size is not None, it must have the same shape as the provided size and must match the type of the output values. Returns ------- out : ndarray or scalar Drawn samples from the parameterized standard gamma distribution. See Also -------- scipy.stats.gamma : probability density function, distribution or cumulative density function, etc. Notes ----- The probability density for the Gamma distribution is .. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)}, where :math:`k` is the shape and :math:`\theta` the scale, and :math:`\Gamma` is the Gamma function. The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant. References ---------- .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GammaDistribution.html .. [2] Wikipedia, "Gamma distribution", https://en.wikipedia.org/wiki/Gamma_distribution Examples -------- Draw samples from the distribution: >>> shape, scale = 2., 1. # mean and width >>> rng = np.random.default_rng() >>> s = rng.standard_gamma(shape, 1000000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> import scipy.special as sps # doctest: +SKIP >>> count, bins, _ = plt.hist(s, 50, density=True) >>> y = bins**(shape-1) * ((np.exp(-bins/scale))/ # doctest: +SKIP ... (sps.gamma(shape) * scale**shape)) >>> plt.plot(bins, y, linewidth=2, color='r') # doctest: +SKIP >>> plt.show() standard_exponential(size=None, dtype=np.float64, method='zig', out=None) Draw samples from the standard exponential distribution. `standard_exponential` is identical to the exponential distribution with a scale parameter of 1. Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. dtype : dtype, optional Desired dtype of the result, only `float64` and `float32` are supported. Byteorder must be native. The default value is np.float64. method : str, optional Either 'inv' or 'zig'. 'inv' uses the default inverse CDF method. 'zig' uses the much faster Ziggurat method of Marsaglia and Tsang. out : ndarray, optional Alternative output array in which to place the result. If size is not None, it must have the same shape as the provided size and must match the type of the output values. Returns ------- out : float or ndarray Drawn samples. Examples -------- Output a 3x8000 array: >>> rng = np.random.default_rng() >>> n = rng.standard_exponential((3, 8000)) standard_cauchy(size=None) Draw samples from a standard Cauchy distribution with mode = 0. Also known as the Lorentz distribution. Parameters ---------- size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. Default is None, in which case a single value is returned. Returns ------- samples : ndarray or scalar The drawn samples. Notes ----- The probability density function for the full Cauchy distribution is .. math:: P(x; x_0, \gamma) = \frac{1}{\pi \gamma \bigl[ 1+ (\frac{x-x_0}{\gamma})^2 \bigr] } and the Standard Cauchy distribution just sets :math:`x_0=0` and :math:`\gamma=1` The Cauchy distribution arises in the solution to the driven harmonic oscillator problem, and also describes spectral line broadening. It also describes the distribution of values at which a line tilted at a random angle will cut the x axis. When studying hypothesis tests that assume normality, seeing how the tests perform on data from a Cauchy distribution is a good indicator of their sensitivity to a heavy-tailed distribution, since the Cauchy looks very much like a Gaussian distribution, but with heavier tails. References ---------- .. [1] NIST/SEMATECH e-Handbook of Statistical Methods, "Cauchy Distribution", https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm .. [2] Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CauchyDistribution.html .. [3] Wikipedia, "Cauchy distribution" https://en.wikipedia.org/wiki/Cauchy_distribution Examples -------- Draw samples and plot the distribution: >>> import matplotlib.pyplot as plt >>> rng = np.random.default_rng() >>> s = rng.standard_cauchy(1000000) >>> s = s[(s>-25) & (s<25)] # truncate distribution so it plots well >>> plt.hist(s, bins=100) >>> plt.show() poisson(lam=1.0, size=None) Draw samples from a Poisson distribution. The Poisson distribution is the limit of the binomial distribution for large N. Parameters ---------- lam : float or array_like of floats Expected number of events occurring in a fixed-time interval, must be >= 0. A sequence must be broadcastable over the requested size. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``lam`` is a scalar. Otherwise, ``np.array(lam).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Poisson distribution. Notes ----- The Poisson distribution .. math:: f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!} For events with an expected separation :math:`\lambda` the Poisson distribution :math:`f(k; \lambda)` describes the probability of :math:`k` events occurring within the observed interval :math:`\lambda`. Because the output is limited to the range of the C int64 type, a ValueError is raised when `lam` is within 10 sigma of the maximum representable value. References ---------- .. [1] Weisstein, Eric W. "Poisson Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PoissonDistribution.html .. [2] Wikipedia, "Poisson distribution", https://en.wikipedia.org/wiki/Poisson_distribution Examples -------- Draw samples from the distribution: >>> rng = np.random.default_rng() >>> lam, size = 5, 10000 >>> s = rng.poisson(lam=lam, size=size) Verify the mean and variance, which should be approximately ``lam``: >>> s.mean(), s.var() (4.9917 5.1088311) # may vary Display the histogram and probability mass function: >>> import matplotlib.pyplot as plt >>> from scipy import stats >>> x = np.arange(0, 21) >>> pmf = stats.poisson.pmf(x, mu=lam) >>> plt.hist(s, bins=x, density=True, width=0.5) >>> plt.stem(x, pmf, 'C1-') >>> plt.show() Draw each 100 values for lambda 100 and 500: >>> s = rng.poisson(lam=(100., 500.), size=(100, 2)) ' object which is not a subclass of 'Sequence'; `shuffle` is not guaranteed to behave correctly. E.g., non-numpy array/tensor objects with view semantics may contain duplicates after shuffling. laplace(loc=0.0, scale=1.0, size=None) Draw samples from the Laplace or double exponential distribution with specified location (or mean) and scale (decay). The Laplace distribution is similar to the Gaussian/normal distribution, but is sharper at the peak and has fatter tails. It represents the difference between two independent, identically distributed exponential random variables. Parameters ---------- loc : float or array_like of floats, optional The position, :math:`\mu`, of the distribution peak. Default is 0. scale : float or array_like of floats, optional :math:`\lambda`, the exponential decay. Default is 1. Must be non- negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n, k)``, then ``m * n * k`` samples are drawn. If size is ``None`` (default), a single value is returned if ``loc`` and ``scale`` are both scalars. Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn. Returns ------- out : ndarray or scalar Drawn samples from the parameterized Laplace distribution. Notes ----- It has the probability density function .. math:: f(x; \mu, \lambda) = \frac{1}{2\lambda} \exp\left(-\frac{|x - \mu|}{\lambda}\right). The first law of Laplace, from 1774, states that the frequency of an error can be expressed as an exponential function of the absolute magnitude of the error, which leads to the Laplace distribution. For many problems in economics and health sciences, this distribution seems to model the data better than the standard Gaussian distribution. References ---------- .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing," New York: Dover, 1972. .. [2] Kotz, Samuel, et. al. "The Laplace Distribution and Generalizations, " Birkhauser, 2001. .. [3] Weisstein, Eric W. "Laplace Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LaplaceDistribution.html .. [4] Wikipedia, "Laplace distribution", https://en.wikipedia.org/wiki/Laplace_distribution Examples -------- Draw samples from the distribution >>> loc, scale = 0., 1. >>> rng = np.random.default_rng() >>> s = rng.laplace(loc, scale, 1000) Display the histogram of the samples, along with the probability density function: >>> import matplotlib.pyplot as plt >>> count, bins, _ = plt.hist(s, 30, density=True) >>> x = np.arange(-8., 8., .01) >>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale) >>> plt.plot(x, pdf) Plot Gaussian for comparison: >>> g = (1/(scale * np.sqrt(2 * np.pi)) * ... np.exp(-(x - loc)**2 / (2 * scale**2))) >>> plt.plot(x,g) dirichlet(alpha, size=None) Draw samples from the Dirichlet distribution. Draw `size` samples of dimension k from a Dirichlet distribution. A Dirichlet-distributed random variable can be seen as a multivariate generalization of a Beta distribution. The Dirichlet distribution is a conjugate prior of a multinomial distribution in Bayesian inference. Parameters ---------- alpha : sequence of floats, length k Parameter of the distribution (length ``k`` for sample of length ``k``). size : int or tuple of ints, optional Output shape. If the given shape is, e.g., ``(m, n)``, then ``m * n * k`` samples are drawn. Default is None, in which case a vector of length ``k`` is returned. Returns ------- samples : ndarray, The drawn samples, of shape ``(size, k)``. Raises ------ ValueError If any value in ``alpha`` is less than zero Notes ----- The Dirichlet distribution is a distribution over vectors :math:`x` that fulfil the conditions :math:`x_i>0` and :math:`\sum_{i=1}^k x_i = 1`. The probability density function :math:`p` of a Dirichlet-distributed random vector :math:`X` is proportional to .. math:: p(x) \propto \prod_{i=1}^{k}{x^{\alpha_i-1}_i}, where :math:`\alpha` is a vector containing the positive concentration parameters. The method uses the following property for computation: let :math:`Y` be a random vector which has components that follow a standard gamma distribution, then :math:`X = \frac{1}{\sum_{i=1}^k{Y_i}} Y` is Dirichlet-distributed References ---------- .. [1] David McKay, "Information Theory, Inference and Learning Algorithms," chapter 23, https://www.inference.org.uk/mackay/itila/ .. [2] Wikipedia, "Dirichlet distribution", https://en.wikipedia.org/wiki/Dirichlet_distribution Examples -------- Taking an example cited in Wikipedia, this distribution can be used if one wanted to cut strings (each of initial length 1.0) into K pieces with different lengths, where each piece had, on average, a designated average length, but allowing some variation in the relative sizes of the pieces. >>> rng = np.random.default_rng() >>> s = rng.dirichlet((10, 5, 3), 20).transpose() >>> import matplotlib.pyplot as plt >>> plt.barh(range(20), s[0]) >>> plt.barh(range(20), s[1], left=s[0], color='g') >>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r') >>> plt.title("Lengths of Strings") Unsupported dtype %r for randomGenerator.triangular (line 2874)Generator.standard_t (line 1871)Generator.standard_exponentialGenerator.noncentral_chisquareGenerator.exponential (line 445)unable to allocate array data.Generator.multivariate_normalGenerator.logseries (line 3637)Generator.lognormal (line 2624)Generator.geometric (line 3427)Generator.dirichlet (line 4427)Generator.chisquare (line 1644)Generator.vonmises (line 1978)Generator.rayleigh (line 2736)Generator.permuted (line 4633)Generator.logistic (line 2541)Generator.binomial (line 2975)numpy/random/_generator.pyxnsample must be nonnegative.multivariate_hypergeometricitemsize <= 0 for cython.arraya and p must have same sizeStep may not be zero (axis %d)Generator.weibull (line 2137)Generator.uniform (line 1009)Generator.shuffle (line 4794)Generator.poisson (line 3256)Generator.negative_binomialGenerator.laplace (line 2336)Generator.integers (line 579)nsample must be an integermean must be 1 dimensionalRange exceeds valid boundsIndex out of bounds (axis %d)Generator.pareto (line 2062)Generator.normal (line 1188)Generator.gumbel (line 2422)out must be a numpy arraynsample must not exceed %dProbabilities contain NaNGenerator.standard_normalGenerator.standard_cauchyGenerator.random (line 301)Generator.power (line 2235)Generator.gamma (line 1385)Generator.choice (line 741)_Generator__generator_ctorDimension %d is not directGenerator.zipf (line 3338)Generator.wald (line 2805)Generator.standard_gammaGenerator.spawn (line 243)Generator.hypergeometricGenerator.bytes (line 706)p must be 1-dimensionalnumpy.random._generatorGenerator.beta (line 364)Generator.noncentral_fCannot index with type 'numpy.lib.array_utilsdefault_rng (line 4999)Invalid shape in axis Generator.permutationGenerator.multinomialGenerator.f (line 1464)Generator.exponentialstandard_exponentialnormalize_axis_indexnoncentral_chisquareGenerator.triangularGenerator.standard_tGenerator.__setstate__Generator.__getstate__you are shuffling a 'multivariate_normalNotImplementedErrorGenerator.logseriesGenerator.lognormalGenerator.geometricGenerator.dirichletGenerator.chisquarensample > sum(colors)ngood + nbad < nsamplecline_in_tracebackasyncio.coroutinesarray is read-onlyGenerator.vonmisesGenerator.rayleighGenerator.permutedGenerator.logisticGenerator.integersGenerator.binomialGenerator.__reduce____pyx_unpickle_Enumnegative_binomialascontiguousarrayGenerator.weibullGenerator.uniformGenerator.shuffleGenerator.poissonGenerator.laplacemay_share_memoryGenerator.randomGenerator.paretoGenerator.normalGenerator.gumbelGenerator.choiceuniform_samplesstandard_normalstandard_cauchy__setstate_cython____pyx_PickleError_poisson_lam_maxdtype_is_objectcollections.abcalpha_csum_dataallocate_bufferView.MemoryViewGenerator.spawnGenerator.powerGenerator.gammaGenerator.bytesunique_indicesstandard_gammainvalid_colorshypergeometric__generator_ctoralpha_csum_arrRuntimeWarningGenerator.zipfGenerator.waldGenerator.betaAssertionError__reduce_cython__count_nonzero__class_getitem__bit_generatorOverflowErrorHYPERGEOM_MAXversion_infovariates_ptrsearchsortedreturn_indexrandoms_data__pyx_checksumnumpy.linalgnum_variatesnoncentral_f_is_coroutine_initializingpermutationmultinomialmax_lam_arrfinal_shapeexponentialdefault_rngcollectionscheck_validUserWarningPickleErrorOutput size MemoryErrorImportErrorGenerator.ftriangularto_shufflestandard_tstacklevelslice_repr__pyx_vtable____pyx_resultpop_size_inum_colorsn_childrenmode > rightlogical_orleft == rightissubdtypeflat_foundempty_likecolors_ptralpha_dataa_originalValueErrorIndexErrorwriteablesum(pvals__reduce_ex____pyx_statemax_indexmarginalslogserieslognormalleft > modeisenabledis_scalarhasobjectgeometricenumeratedirichletchisquarealpha_arrTypeErrorGeneratorwarningsvonmisesvariatesval_datatemp_arrswapaxessubtract__setstate__set_sizereversedregister__reduce__rayleigh__pyx_typepop_size_picklepermutedoperatoronsamplen_uint32logisticlnsampleitemsizeisscalarisnativeisfiniteintegersidx_datahigh - 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0A(B BBBI !\!XH!T8BBB E(A0A8DP1 8D0A(B BBBA "H("DX<"@ BIA A(G`D (D ABBG DhWp_hF`b (D ABBE ("BAA Q ABH X"htBIB B(A0A8GzV_F\ 8C0A(B BBBK X #tBIB B(A0A8GzV_F\ 8C0A(B BBBK X|#tBIB B(A0A8GzV_F\ 8C0A(B BBBK $#Ԝ BIB B(D0D8GGGGGGGGGGGGGGGGGGGGGGGGGGS 8D0A(B BBBA AGGGGGGGGGGGGGGGGGGGGGGGGGS % BBB E(D0D8GTGGGGGGGGGGGGGGGGGGGGGGGGGGGG G G G G G G G G G G G G G G G G G G G G G G LW 8A0A(B BBBF p$&2BIB E(D0A8D 8D0A(B BBBJ DW_F^ 8A0A(B BBBH d&BBB B(A0A8D@ 8A0A(B BBBD j 8F0A(B BBBG D'BBA  DBD ,(R0`(C A BBB DH'DBBA  DBD ,(R0`(C A BBB D'BBA  DBD ,(R0`(C A BBB 'BIB B(D0A8G 8A0A(B BBBA DV_F\ 8C0A(B BBBI  8C0A(B BBBJ dh(4BEB B(D0A8D 8A0A(B BBBB  8A0A(B BBBH H(BBB B(A0A8DPN 8D0A(B BBBA )=l0)<BBB D(A0D@[ 0A(A BBBF V 0A(A BBBJ  0C(A BBBJ )lBIG B(D0O8PDdBFFBFABFQW 8D0A(B BBBE VRF,*`BIG B(D0O8PDdBFFBFABFQW 8D0A(B BBBE VRF\*TM BIB B(A0D8GV_F_ 8D0A(B BBBE +DD ~ AA <+ %BIB B(D0A8GL`F_ 8D0A(B BBBH drBAABADBFQ+%BIB B(D0A8GL`F_ 8D0A(B BBBH drBAABADBFQL,p DBIB B(D0A8GL`F_ 8D0A(B BBBF ldBABBGEFAQH,8BBE B(A0A8DP 8D0A(B BBBJ d -W BBB E(D0A8Dp 8A0A(B BBBB  8A0A(B BBBB -dDZ B CH-BBE B(D0A8Gp9 8D0A(B BBBC -$Dj B C E L.%BBE E(A0A8Dn 8D0A(B BBBA Hh.t-BEB B(A0A8D@ 8D0A(B BBBE `.2IBB A(D0 (D BBBB H (D BBBF I (D BBBE /5]BIB B(D0A8G^gBABBABBAQvHYF_ 8D0A(B BBBI /9 BMQ F(D0A8D+WEFFBFABFQ[ 8D0A(B BBBD WRFW_A@0C BMQ H(D0A8D)WEFFBFABFQ[ 8D0A(B BBBD WRFW_A0M BMQ H(D0A8D)WEFFBFABFQ[ 8D0A(B BBBD WRFW_Ax1W BMQ H(D0A8D)WEFFBFABFQ[ 8D0A(B BBBD WRFW_A2a BMN H(D0A8D,WEFFBFABFQ[ 8D0A(B BBBD WRFW_A2kBMG B(D0L8D 8A0A(B BBBJ dW_F^ 8A0A(B BBBH W_A 8A0A(B BBBH V`Gd3}BIB B(D0A8GORF_ 8D0A(B BBBE KrBAABADBFQ3,BIB B(D0A8GORF_ 8D0A(B BBBE KrBAABADBFQ|4PBIB B(D0A8GORF_ 8D0A(B BBBE KrBAABADBFQ5tBIB B(D0A8GORF_ 8D0A(B BBBE SdBABBGEFAQ5BIB B(D0A8GORF_ 8D0A(B BBBE SdBABBGEFAQ 6BIB B(D0A8GORF_ 8D0A(B BBBE KrBAABADBFQ6BIB B(D0A8GORF_ 8D0A(B BBBE SdBABBGEFAQ87BIB B(D0A8GORF_ 8D0A(B BBBE KrBAABADBFQ7( BIG B(L0F8KdBFFBFABFQ[ 8D0A(B BBBI VKFLP8BEB E(A0A8D 8D0A(B BBBG @8l!bBB A(A0D@S 0D(A BBBF $8WBDA LABH 9*BBE B(A0A8DP8D0A(B BBB@X9BBE A(A0D@z 0D(A BBBD H90tBHA D(G0s (D ABBF N(D ABBX9d[#BIE E(A0A8DPRF_ 8D0A(B BBBD XD:h*BIB E(A0A8GPRFb 8D0A(B BBBE :&KCBIB B(A0A8Jz VRFb 8D0A(B BBBF W`F GIB!GNA8;itBMG B(A0K8JK VRFa 8D0A(B BBBK  V`GIRA;y$BIB B(A0A8JVRFb 8D0A(B BBBI MeFABFBBRgLLEBFABFBY!W_A)V`G<`"BIB B(A0A8J eBBBBABTgVRFb 8D0A(B BBBA IBKHBABBAYW_AIV`Gh=#`BIE B(A0A8GtBAABADBFMWKFb 8D0A(B BBBI BGA\>3TBIG E(A0A8ZWRF_ 8D0A(B BBBD h>CBIG B(D0A8ZrBAFBFBBAMFMJFBFBGARWRF_ 8D0A(B BBBD ? ^ BBB B(D0D8G& 8D0A(B BBBE _HPB=EG^AhLBl?n\ BIQ N(E0Q8ZEJIAG 8A0A(B BBBD MORFt@z BIE B(D0A8D!W_F^ 8A0A(B BBBH   8C0A(B BBBD @&+BIG B(A0Q8c9YQAk 8D0A(B BBBK QYTAEORFYRBYRB\QB BQABRDERBDQA\AбTBIG E(A0A8ZWRF_ 8D0A(B BBBD XA4 BBB B(A0A8DP 8D0A(B BBBA aXN`gXAP HBBBB B(A0A8GbHHK/BFFAAAAACNREFFAAAAACNYEFFAAAAACNYEFFAAAAACN`EFFAAAAACNQTbAoEFFAAAAACNgEFFAAAAACNEFFAAAAACNxTcBoEFFAAAAACNV[MMPoEFFAAAAACN`EFFAAAAACNQMFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFMhoEFFAAAAACNDTTMMFFFIoEFFAAAAACNEFFAAAAACNgEFFAAAAACNV[\BoEFFAAAAACNgEFFAAAAACNgEFFAAAAACNnEFFAAAAACN`EFFAAAAACNgEFFAAAAACNYEFFAAAAACNtEFFAAFAAAKgEFFAAAAACN`EFFAAAAACNtEFFAAFAAAKtEFFAAFAAAKtEFFAAFAAAKtEFFAAFAAAKtEFFAAFAAAKgEFFAAAAACNtEFFAAFAAAKgEFFAAAAACNRTTMMFFFIoEFFAAAAACNRTTMMFFFFFIoEFFAAAAACNV[MMMFIoEFFAAAAACN`EFFAAAAACNtEFFAAFAAAK`EFFAAAAACNV[MMMFFFIoEFFAAAAACNtEFFAAFAAAKV[MMFFFFFFFFFFPoEFFAAAAAFNEMFFFFFFFFFFFFFFFMMMZoEFFAAAAACNQTTMMFFFFFFFIoEFFAAAAACN|[MMFFFFFFFFFFPoEFFAAAAACNV[MMMFFFIoEFFAAAAAFNV[MMMFIoEFFAAAAACNtTbAoEFFAAAAACNREFFAAAAACN_8A0A(B BBB\N BIB B(A0A8G$HYF_ 8D0A(B BBBA |dNBBB E(A0D8G WRFm 8D0A(B BBBB  W_FVaG NBLB B(D0A8Jq 8A0A(B BBBA GGGGGGGGGGGGGGGGGGGGGGGGGFFFFFFFFFFCCCCCCCCCCCCCCCBS@PWAJvLKA AF d CA ,LP|AQ N DR `AC F H|P0IBAJvLKA ABH l CBA ,P4|AQ N DR `AC F \P>-BIG B(A0A8G Y_Fb 8D0A(B BBBG XQd$ BIB E(A0A8J!XBFABFAEFQ^ 8D0A(B BBBH zVRF(W_A9V`GXR04DBIB E(A0A8JPRFb 8D0A(B BBBG dRtWBMG B(A0Q8JVRFb 8D0A(B BBBB H^AKW`GRBMG B(D0A8QWRF_ 8D0A(B BBBD gFFBFBBFAO_W_F\SBIB B(A0A8J? 8D0A(B BBBI L`FlS+BMG E(A0A8Tx WRFb 8D0A(B BBBI  V`GTT& BIG B(D0O8PdkBAFBFABATW 8D0A(B BBBE VRFT 1 BIG B(D0O8PddBFFBFABFQW 8D0A(B BBBE VRFlU0; BIG B(D0O8PddBFFBFABFQW 8D0A(B BBBE VRFUJ  V@E%G]$VXE 88VTEIGED D(F0a(A ABBG8tVhEYGED D(F0t(A ABBDHVEBEH H(KP (E ABBK [(A AFB8V`FIGED D(F0b(A ABBFT8WtF-BBE H(H0K@ 0D(A BBBE m0A(A FBB8WLGIGED D(F0b(A ABBF8W`GaGED D(F0v(A ABBJ<XGyGED D(G0O(A ABBHHHXGBBE I(H0K8K` 8E0A(B BBBE 8XIIGED D(F0b(A ABBFHX,IBBE B(H0H8KP 8F0A(B BBBD 8YJIGED D(F0b(A ABBFHXYJAG@ AI t AK 1 AF J EI J AE HYMAG0 AC q AF B AE J EA J AE Y|OGJZOGJ ZOGJ8ZO 0LZOvAD@O EAH `Z,Q$D _ZDQD UZLQ(D cZdQD UZlQD U0ZtQAG@ AO t EA A,[@SDQD[HSSAG }Ad[S/AG ]A[SD U[SB\ ][SCD z([TAG V AI XA$[tTiBFG0IAK \TNAG DA@\TDIX\T3D g A t\UAG _ AE l\UBFB A(A0Gpg 0D(A BBBK V 0D(A BBBG / 0D(A BBBA ]DX&IXL$]XX BFG E(D0A8J 8A0A(E BBBF 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