# Python: 3.11.9 (tags/v3.11.9:de54cf5, Apr 2 2024, 10:12:12) [MSC v.1938 64 bit (AMD64)] # Library: numpy, version: 1.26.4 # Module: numpy.random._generator, version: unspecified import typing import builtins as _mod_builtins import collections.abc as _mod_collections_abc import numpy.random._pcg64 as _mod_numpy_random__pcg64 class Generator(_mod_builtins.object): "\n Generator(bit_generator)\n\n Container for the BitGenerators.\n\n ``Generator`` exposes a number of methods for generating random\n numbers drawn from a variety of probability distributions. In addition to\n the distribution-specific arguments, each method takes a keyword argument\n `size` that defaults to ``None``. If `size` is ``None``, then a single\n value is generated and returned. If `size` is an integer, then a 1-D\n array filled with generated values is returned. If `size` is a tuple,\n then an array with that shape is filled and returned.\n\n The function :func:`numpy.random.default_rng` will instantiate\n a `Generator` with numpy's default `BitGenerator`.\n\n **No Compatibility Guarantee**\n\n ``Generator`` does not provide a version compatibility guarantee. In\n particular, as better algorithms evolve the bit stream may change.\n\n Parameters\n ----------\n bit_generator : BitGenerator\n BitGenerator to use as the core generator.\n\n Notes\n -----\n The Python stdlib module `random` contains pseudo-random number generator\n with a number of methods that are similar to the ones available in\n ``Generator``. It uses Mersenne Twister, and this bit generator can\n be accessed using ``MT19937``. ``Generator``, besides being\n NumPy-aware, has the advantage that it provides a much larger number\n of probability distributions to choose from.\n\n Examples\n --------\n >>> from numpy.random import Generator, PCG64\n >>> rng = Generator(PCG64())\n >>> rng.standard_normal()\n -0.203 # random\n\n See Also\n --------\n default_rng : Recommended constructor for `Generator`.\n " def __getstate__(self) -> typing.Any: ... def __init__(self, bit_generator) -> None: "\n Generator(bit_generator)\n\n Container for the BitGenerators.\n\n ``Generator`` exposes a number of methods for generating random\n numbers drawn from a variety of probability distributions. In addition to\n the distribution-specific arguments, each method takes a keyword argument\n `size` that defaults to ``None``. If `size` is ``None``, then a single\n value is generated and returned. If `size` is an integer, then a 1-D\n array filled with generated values is returned. If `size` is a tuple,\n then an array with that shape is filled and returned.\n\n The function :func:`numpy.random.default_rng` will instantiate\n a `Generator` with numpy's default `BitGenerator`.\n\n **No Compatibility Guarantee**\n\n ``Generator`` does not provide a version compatibility guarantee. In\n particular, as better algorithms evolve the bit stream may change.\n\n Parameters\n ----------\n bit_generator : BitGenerator\n BitGenerator to use as the core generator.\n\n Notes\n -----\n The Python stdlib module `random` contains pseudo-random number generator\n with a number of methods that are similar to the ones available in\n ``Generator``. It uses Mersenne Twister, and this bit generator can\n be accessed using ``MT19937``. ``Generator``, besides being\n NumPy-aware, has the advantage that it provides a much larger number\n of probability distributions to choose from.\n\n Examples\n --------\n >>> from numpy.random import Generator, PCG64\n >>> rng = Generator(PCG64())\n >>> rng.standard_normal()\n -0.203 # random\n\n See Also\n --------\n default_rng : Recommended constructor for `Generator`.\n " ... @classmethod def __init_subclass__(cls) -> None: 'This method is called when a class is subclassed.\n\nThe default implementation does nothing. It may be\noverridden to extend subclasses.\n' ... def __reduce__(self) -> typing.Union[str, typing.Tuple[typing.Any, ...]]: ... def __repr__(self) -> str: ... def __setstate__(self, state: typing.Any) -> None: ... def __str__(self) -> str: 'Return str(self).' ... @classmethod def __subclasshook__(cls, subclass: typing.Any) -> bool: 'Abstract classes can override this to customize issubclass().\n\nThis is invoked early on by abc.ABCMeta.__subclasscheck__().\nIt should return True, False or NotImplemented. If it returns\nNotImplemented, the normal algorithm is used. Otherwise, it\noverrides the normal algorithm (and the outcome is cached).\n' ... @property def _bit_generator(self) -> typing.Any: ... _poisson_lam_max: float def beta(self, a, b, size=...) -> typing.Any: ' Draw samples from a Beta distribution.\n\n The Beta distribution is a special case of the Dirichlet distribution,\n and is related to the Gamma distribution. It has the probability\n distribution function\n\n .. math:: f(x; a,b) = \\frac{1}{B(\\alpha, \\beta)} x^{\\alpha - 1}\n (1 - x)^{\\beta - 1},\n\n where the normalization, B, is the beta function,\n\n .. math:: B(\\alpha, \\beta) = \\int_0^1 t^{\\alpha - 1}\n (1 - t)^{\\beta - 1} dt.\n\n It is often seen in Bayesian inference and order statistics.\n\n Parameters\n ----------\n a : float or array_like of floats\n Alpha, positive (>0).\n b : float or array_like of floats\n Beta, positive (>0).\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``a`` and ``b`` are both scalars.\n Otherwise, ``np.broadcast(a, b).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized beta distribution.\n\n ' ... def binomial(self, n, p, size=...) -> typing.Any: ' Draw samples from a binomial distribution.\n\n Samples are drawn from a binomial distribution with specified\n parameters, n trials and p probability of success where\n n an integer >= 0 and p is in the interval [0,1]. (n may be\n input as a float, but it is truncated to an integer in use)\n\n Parameters\n ----------\n n : int or array_like of ints\n Parameter of the distribution, >= 0. Floats are also accepted,\n but they will be truncated to integers.\n p : float or array_like of floats\n Parameter of the distribution, >= 0 and <=1.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``n`` and ``p`` are both scalars.\n Otherwise, ``np.broadcast(n, p).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized binomial distribution, where\n each sample is equal to the number of successes over the n trials.\n\n See Also\n --------\n scipy.stats.binom : probability density function, distribution or\n cumulative density function, etc.\n\n Notes\n -----\n The probability density for the binomial distribution is\n\n .. math:: P(N) = \\binom{n}{N}p^N(1-p)^{n-N},\n\n where :math:`n` is the number of trials, :math:`p` is the probability\n of success, and :math:`N` is the number of successes.\n\n When estimating the standard error of a proportion in a population by\n using a random sample, the normal distribution works well unless the\n product p*n <=5, where p = population proportion estimate, and n =\n number of samples, in which case the binomial distribution is used\n instead. For example, a sample of 15 people shows 4 who are left\n handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4,\n so the binomial distribution should be used in this case.\n\n References\n ----------\n .. [1] Dalgaard, Peter, "Introductory Statistics with R",\n Springer-Verlag, 2002.\n .. [2] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill,\n Fifth Edition, 2002.\n .. [3] Lentner, Marvin, "Elementary Applied Statistics", Bogden\n and Quigley, 1972.\n .. [4] Weisstein, Eric W. "Binomial Distribution." From MathWorld--A\n Wolfram Web Resource.\n http://mathworld.wolfram.com/BinomialDistribution.html\n .. [5] Wikipedia, "Binomial distribution",\n https://en.wikipedia.org/wiki/Binomial_distribution\n\n Examples\n --------\n Draw samples from the distribution:\n\n >>> rng = np.random.default_rng()\n >>> n, p = 10, .5 # number of trials, probability of each trial\n >>> s = rng.binomial(n, p, 1000)\n # result of flipping a coin 10 times, tested 1000 times.\n\n A real world example. A company drills 9 wild-cat oil exploration\n wells, each with an estimated probability of success of 0.1. All nine\n wells fail. What is the probability of that happening?\n\n Let\'s do 20,000 trials of the model, and count the number that\n generate zero positive results.\n\n >>> sum(rng.binomial(9, 0.1, 20000) == 0)/20000.\n # answer = 0.38885, or 39%.\n\n ' ... @property def bit_generator(self) -> typing.Any: '\n Gets the bit generator instance used by the generator\n\n Returns\n -------\n bit_generator : BitGenerator\n The bit generator instance used by the generator\n ' ... def bytes(self, length) -> typing.Any: " Return random bytes.\n\n Parameters\n ----------\n length : int\n Number of random bytes.\n\n Returns\n -------\n out : bytes\n String of length `length`.\n\n Examples\n --------\n >>> np.random.default_rng().bytes(10)\n b'\\xfeC\\x9b\\x86\\x17\\xf2\\xa1\\xafcp' # random\n\n " ... def chisquare(self, df, size=...) -> typing.Any: ' Draw samples from a chi-square distribution.\n\n When `df` independent random variables, each with standard normal\n distributions (mean 0, variance 1), are squared and summed, the\n resulting distribution is chi-square (see Notes). This distribution\n is often used in hypothesis testing.\n\n Parameters\n ----------\n df : float or array_like of floats\n Number of degrees of freedom, must be > 0.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``df`` is a scalar. Otherwise,\n ``np.array(df).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized chi-square distribution.\n\n Raises\n ------\n ValueError\n When `df` <= 0 or when an inappropriate `size` (e.g. ``size=-1``)\n is given.\n\n Notes\n -----\n The variable obtained by summing the squares of `df` independent,\n standard normally distributed random variables:\n\n .. math:: Q = \\sum_{i=0}^{\\mathtt{df}} X^2_i\n\n is chi-square distributed, denoted\n\n .. math:: Q \\sim \\chi^2_k.\n\n The probability density function of the chi-squared distribution is\n\n .. math:: p(x) = \\frac{(1/2)^{k/2}}{\\Gamma(k/2)}\n x^{k/2 - 1} e^{-x/2},\n\n where :math:`\\Gamma` is the gamma function,\n\n .. math:: \\Gamma(x) = \\int_0^{-\\infty} t^{x - 1} e^{-t} dt.\n\n References\n ----------\n .. [1] NIST "Engineering Statistics Handbook"\n https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm\n\n Examples\n --------\n >>> np.random.default_rng().chisquare(2,4)\n array([ 1.89920014, 9.00867716, 3.13710533, 5.62318272]) # random\n\n ' ... def choice(self, a, size=..., replace=..., p=..., axis=..., shuffle=...) -> typing.Any: " Generates a random sample from a given array\n\n Parameters\n ----------\n a : {array_like, int}\n If an ndarray, a random sample is generated from its elements.\n If an int, the random sample is generated from np.arange(a).\n size : {int, tuple[int]}, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn from the 1-d `a`. If `a` has more\n than one dimension, the `size` shape will be inserted into the\n `axis` dimension, so the output ``ndim`` will be ``a.ndim - 1 +\n len(size)``. Default is None, in which case a single value is\n returned.\n replace : bool, optional\n Whether the sample is with or without replacement. Default is True,\n meaning that a value of ``a`` can be selected multiple times.\n p : 1-D array_like, optional\n The probabilities associated with each entry in a.\n If not given, the sample assumes a uniform distribution over all\n entries in ``a``.\n axis : int, optional\n The axis along which the selection is performed. The default, 0,\n selects by row.\n shuffle : bool, optional\n Whether the sample is shuffled when sampling without replacement.\n Default is True, False provides a speedup.\n\n Returns\n -------\n samples : single item or ndarray\n The generated random samples\n\n Raises\n ------\n ValueError\n If a is an int and less than zero, if p is not 1-dimensional, if\n a is array-like with a size 0, if p is not a vector of\n probabilities, if a and p have different lengths, or if\n replace=False and the sample size is greater than the population\n size.\n\n See Also\n --------\n integers, shuffle, permutation\n\n Notes\n -----\n Setting user-specified probabilities through ``p`` uses a more general but less\n efficient sampler than the default. The general sampler produces a different sample\n than the optimized sampler even if each element of ``p`` is 1 / len(a).\n\n Examples\n --------\n Generate a uniform random sample from np.arange(5) of size 3:\n\n >>> rng = np.random.default_rng()\n >>> rng.choice(5, 3)\n array([0, 3, 4]) # random\n >>> #This is equivalent to rng.integers(0,5,3)\n\n Generate a non-uniform random sample from np.arange(5) of size 3:\n\n >>> rng.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0])\n array([3, 3, 0]) # random\n\n Generate a uniform random sample from np.arange(5) of size 3 without\n replacement:\n\n >>> rng.choice(5, 3, replace=False)\n array([3,1,0]) # random\n >>> #This is equivalent to rng.permutation(np.arange(5))[:3]\n\n Generate a uniform random sample from a 2-D array along the first\n axis (the default), without replacement:\n\n >>> rng.choice([[0, 1, 2], [3, 4, 5], [6, 7, 8]], 2, replace=False)\n array([[3, 4, 5], # random\n [0, 1, 2]])\n\n Generate a non-uniform random sample from np.arange(5) of size\n 3 without replacement:\n\n >>> rng.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0])\n array([2, 3, 0]) # random\n\n Any of the above can be repeated with an arbitrary array-like\n instead of just integers. For instance:\n\n >>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher']\n >>> rng.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3])\n array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], # random\n dtype=' typing.Any: ' Draw samples from the Dirichlet distribution.\n\n Draw `size` samples of dimension k from a Dirichlet distribution. A\n Dirichlet-distributed random variable can be seen as a multivariate\n generalization of a Beta distribution. The Dirichlet distribution\n is a conjugate prior of a multinomial distribution in Bayesian\n inference.\n\n Parameters\n ----------\n alpha : sequence of floats, length k\n Parameter of the distribution (length ``k`` for sample of\n length ``k``).\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n)``, then\n ``m * n * k`` samples are drawn. Default is None, in which case a\n vector of length ``k`` is returned.\n\n Returns\n -------\n samples : ndarray,\n The drawn samples, of shape ``(size, k)``.\n\n Raises\n ------\n ValueError\n If any value in ``alpha`` is less than zero\n\n Notes\n -----\n The Dirichlet distribution is a distribution over vectors\n :math:`x` that fulfil the conditions :math:`x_i>0` and\n :math:`\\sum_{i=1}^k x_i = 1`.\n\n The probability density function :math:`p` of a\n Dirichlet-distributed random vector :math:`X` is\n proportional to\n\n .. math:: p(x) \\propto \\prod_{i=1}^{k}{x^{\\alpha_i-1}_i},\n\n where :math:`\\alpha` is a vector containing the positive\n concentration parameters.\n\n The method uses the following property for computation: let :math:`Y`\n be a random vector which has components that follow a standard gamma\n distribution, then :math:`X = \\frac{1}{\\sum_{i=1}^k{Y_i}} Y`\n is Dirichlet-distributed\n\n References\n ----------\n .. [1] David McKay, "Information Theory, Inference and Learning\n Algorithms," chapter 23,\n http://www.inference.org.uk/mackay/itila/\n .. [2] Wikipedia, "Dirichlet distribution",\n https://en.wikipedia.org/wiki/Dirichlet_distribution\n\n Examples\n --------\n Taking an example cited in Wikipedia, this distribution can be used if\n one wanted to cut strings (each of initial length 1.0) into K pieces\n with different lengths, where each piece had, on average, a designated\n average length, but allowing some variation in the relative sizes of\n the pieces.\n\n >>> s = np.random.default_rng().dirichlet((10, 5, 3), 20).transpose()\n\n >>> import matplotlib.pyplot as plt\n >>> plt.barh(range(20), s[0])\n >>> plt.barh(range(20), s[1], left=s[0], color=\'g\')\n >>> plt.barh(range(20), s[2], left=s[0]+s[1], color=\'r\')\n >>> plt.title("Lengths of Strings")\n\n ' ... def exponential(self, scale=..., size=...) -> typing.Any: ' Draw samples from an exponential distribution.\n\n Its probability density function is\n\n .. math:: f(x; \\frac{1}{\\beta}) = \\frac{1}{\\beta} \\exp(-\\frac{x}{\\beta}),\n\n for ``x > 0`` and 0 elsewhere. :math:`\\beta` is the scale parameter,\n which is the inverse of the rate parameter :math:`\\lambda = 1/\\beta`.\n The rate parameter is an alternative, widely used parameterization\n of the exponential distribution [3]_.\n\n The exponential distribution is a continuous analogue of the\n geometric distribution. It describes many common situations, such as\n the size of raindrops measured over many rainstorms [1]_, or the time\n between page requests to Wikipedia [2]_.\n\n Parameters\n ----------\n scale : float or array_like of floats\n The scale parameter, :math:`\\beta = 1/\\lambda`. Must be\n non-negative.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``scale`` is a scalar. Otherwise,\n ``np.array(scale).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized exponential distribution.\n\n Examples\n --------\n A real world example: Assume a company has 10000 customer support \n agents and the average time between customer calls is 4 minutes.\n\n >>> n = 10000\n >>> time_between_calls = np.random.default_rng().exponential(scale=4, size=n)\n\n What is the probability that a customer will call in the next \n 4 to 5 minutes? \n \n >>> x = ((time_between_calls < 5).sum())/n \n >>> y = ((time_between_calls < 4).sum())/n\n >>> x-y\n 0.08 # may vary\n\n References\n ----------\n .. [1] Peyton Z. Peebles Jr., "Probability, Random Variables and\n Random Signal Principles", 4th ed, 2001, p. 57.\n .. [2] Wikipedia, "Poisson process",\n https://en.wikipedia.org/wiki/Poisson_process\n .. [3] Wikipedia, "Exponential distribution",\n https://en.wikipedia.org/wiki/Exponential_distribution\n\n ' ... def f(self, dfnum, dfden, size=...) -> typing.Any: ' Draw samples from an F distribution.\n\n Samples are drawn from an F distribution with specified parameters,\n `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of\n freedom in denominator), where both parameters must be greater than\n zero.\n\n The random variate of the F distribution (also known as the\n Fisher distribution) is a continuous probability distribution\n that arises in ANOVA tests, and is the ratio of two chi-square\n variates.\n\n Parameters\n ----------\n dfnum : float or array_like of floats\n Degrees of freedom in numerator, must be > 0.\n dfden : float or array_like of float\n Degrees of freedom in denominator, must be > 0.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``dfnum`` and ``dfden`` are both scalars.\n Otherwise, ``np.broadcast(dfnum, dfden).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized Fisher distribution.\n\n See Also\n --------\n scipy.stats.f : probability density function, distribution or\n cumulative density function, etc.\n\n Notes\n -----\n The F statistic is used to compare in-group variances to between-group\n variances. Calculating the distribution depends on the sampling, and\n so it is a function of the respective degrees of freedom in the\n problem. The variable `dfnum` is the number of samples minus one, the\n between-groups degrees of freedom, while `dfden` is the within-groups\n degrees of freedom, the sum of the number of samples in each group\n minus the number of groups.\n\n References\n ----------\n .. [1] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill,\n Fifth Edition, 2002.\n .. [2] Wikipedia, "F-distribution",\n https://en.wikipedia.org/wiki/F-distribution\n\n Examples\n --------\n An example from Glantz[1], pp 47-40:\n\n Two groups, children of diabetics (25 people) and children from people\n without diabetes (25 controls). Fasting blood glucose was measured,\n case group had a mean value of 86.1, controls had a mean value of\n 82.2. Standard deviations were 2.09 and 2.49 respectively. Are these\n data consistent with the null hypothesis that the parents diabetic\n status does not affect their children\'s blood glucose levels?\n Calculating the F statistic from the data gives a value of 36.01.\n\n Draw samples from the distribution:\n\n >>> dfnum = 1. # between group degrees of freedom\n >>> dfden = 48. # within groups degrees of freedom\n >>> s = np.random.default_rng().f(dfnum, dfden, 1000)\n\n The lower bound for the top 1% of the samples is :\n\n >>> np.sort(s)[-10]\n 7.61988120985 # random\n\n So there is about a 1% chance that the F statistic will exceed 7.62,\n the measured value is 36, so the null hypothesis is rejected at the 1%\n level.\n\n ' ... def gamma(self, shape, scale=..., size=...) -> typing.Any: ' Draw samples from a Gamma distribution.\n\n Samples are drawn from a Gamma distribution with specified parameters,\n `shape` (sometimes designated "k") and `scale` (sometimes designated\n "theta"), where both parameters are > 0.\n\n Parameters\n ----------\n shape : float or array_like of floats\n The shape of the gamma distribution. Must be non-negative.\n scale : float or array_like of floats, optional\n The scale of the gamma distribution. Must be non-negative.\n Default is equal to 1.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``shape`` and ``scale`` are both scalars.\n Otherwise, ``np.broadcast(shape, scale).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized gamma distribution.\n\n See Also\n --------\n scipy.stats.gamma : probability density function, distribution or\n cumulative density function, etc.\n\n Notes\n -----\n The probability density for the Gamma distribution is\n\n .. math:: p(x) = x^{k-1}\\frac{e^{-x/\\theta}}{\\theta^k\\Gamma(k)},\n\n where :math:`k` is the shape and :math:`\\theta` the scale,\n and :math:`\\Gamma` is the Gamma function.\n\n The Gamma distribution is often used to model the times to failure of\n electronic components, and arises naturally in processes for which the\n waiting times between Poisson distributed events are relevant.\n\n References\n ----------\n .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A\n Wolfram Web Resource.\n http://mathworld.wolfram.com/GammaDistribution.html\n .. [2] Wikipedia, "Gamma distribution",\n https://en.wikipedia.org/wiki/Gamma_distribution\n\n Examples\n --------\n Draw samples from the distribution:\n\n >>> shape, scale = 2., 2. # mean=4, std=2*sqrt(2)\n >>> s = np.random.default_rng().gamma(shape, scale, 1000)\n\n Display the histogram of the samples, along with\n the probability density function:\n\n >>> import matplotlib.pyplot as plt\n >>> import scipy.special as sps # doctest: +SKIP\n >>> count, bins, ignored = plt.hist(s, 50, density=True)\n >>> y = bins**(shape-1)*(np.exp(-bins/scale) / # doctest: +SKIP\n ... (sps.gamma(shape)*scale**shape))\n >>> plt.plot(bins, y, linewidth=2, color=\'r\') # doctest: +SKIP\n >>> plt.show()\n\n ' ... def geometric(self, p, size=...) -> typing.Any: ' Draw samples from the geometric distribution.\n\n Bernoulli trials are experiments with one of two outcomes:\n success or failure (an example of such an experiment is flipping\n a coin). The geometric distribution models the number of trials\n that must be run in order to achieve success. It is therefore\n supported on the positive integers, ``k = 1, 2, ...``.\n\n The probability mass function of the geometric distribution is\n\n .. math:: f(k) = (1 - p)^{k - 1} p\n\n where `p` is the probability of success of an individual trial.\n\n Parameters\n ----------\n p : float or array_like of floats\n The probability of success of an individual trial.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``p`` is a scalar. Otherwise,\n ``np.array(p).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized geometric distribution.\n\n Examples\n --------\n Draw ten thousand values from the geometric distribution,\n with the probability of an individual success equal to 0.35:\n\n >>> z = np.random.default_rng().geometric(p=0.35, size=10000)\n\n How many trials succeeded after a single run?\n\n >>> (z == 1).sum() / 10000.\n 0.34889999999999999 # random\n\n ' ... def gumbel(self, loc=..., scale=..., size=...) -> typing.Any: ' Draw samples from a Gumbel distribution.\n\n Draw samples from a Gumbel distribution with specified location and\n scale. For more information on the Gumbel distribution, see\n Notes and References below.\n\n Parameters\n ----------\n loc : float or array_like of floats, optional\n The location of the mode of the distribution. Default is 0.\n scale : float or array_like of floats, optional\n The scale parameter of the distribution. Default is 1. Must be non-\n negative.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``loc`` and ``scale`` are both scalars.\n Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized Gumbel distribution.\n\n See Also\n --------\n scipy.stats.gumbel_l\n scipy.stats.gumbel_r\n scipy.stats.genextreme\n weibull\n\n Notes\n -----\n The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme\n Value Type I) distribution is one of a class of Generalized Extreme\n Value (GEV) distributions used in modeling extreme value problems.\n The Gumbel is a special case of the Extreme Value Type I distribution\n for maximums from distributions with "exponential-like" tails.\n\n The probability density for the Gumbel distribution is\n\n .. math:: p(x) = \\frac{e^{-(x - \\mu)/ \\beta}}{\\beta} e^{ -e^{-(x - \\mu)/\n \\beta}},\n\n where :math:`\\mu` is the mode, a location parameter, and\n :math:`\\beta` is the scale parameter.\n\n The Gumbel (named for German mathematician Emil Julius Gumbel) was used\n very early in the hydrology literature, for modeling the occurrence of\n flood events. It is also used for modeling maximum wind speed and\n rainfall rates. It is a "fat-tailed" distribution - the probability of\n an event in the tail of the distribution is larger than if one used a\n Gaussian, hence the surprisingly frequent occurrence of 100-year\n floods. Floods were initially modeled as a Gaussian process, which\n underestimated the frequency of extreme events.\n\n It is one of a class of extreme value distributions, the Generalized\n Extreme Value (GEV) distributions, which also includes the Weibull and\n Frechet.\n\n The function has a mean of :math:`\\mu + 0.57721\\beta` and a variance\n of :math:`\\frac{\\pi^2}{6}\\beta^2`.\n\n References\n ----------\n .. [1] Gumbel, E. J., "Statistics of Extremes,"\n New York: Columbia University Press, 1958.\n .. [2] Reiss, R.-D. and Thomas, M., "Statistical Analysis of Extreme\n Values from Insurance, Finance, Hydrology and Other Fields,"\n Basel: Birkhauser Verlag, 2001.\n\n Examples\n --------\n Draw samples from the distribution:\n\n >>> rng = np.random.default_rng()\n >>> mu, beta = 0, 0.1 # location and scale\n >>> s = rng.gumbel(mu, beta, 1000)\n\n Display the histogram of the samples, along with\n the probability density function:\n\n >>> import matplotlib.pyplot as plt\n >>> count, bins, ignored = plt.hist(s, 30, density=True)\n >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)\n ... * np.exp( -np.exp( -(bins - mu) /beta) ),\n ... linewidth=2, color=\'r\')\n >>> plt.show()\n\n Show how an extreme value distribution can arise from a Gaussian process\n and compare to a Gaussian:\n\n >>> means = []\n >>> maxima = []\n >>> for i in range(0,1000) :\n ... a = rng.normal(mu, beta, 1000)\n ... means.append(a.mean())\n ... maxima.append(a.max())\n >>> count, bins, ignored = plt.hist(maxima, 30, density=True)\n >>> beta = np.std(maxima) * np.sqrt(6) / np.pi\n >>> mu = np.mean(maxima) - 0.57721*beta\n >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)\n ... * np.exp(-np.exp(-(bins - mu)/beta)),\n ... linewidth=2, color=\'r\')\n >>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi))\n ... * np.exp(-(bins - mu)**2 / (2 * beta**2)),\n ... linewidth=2, color=\'g\')\n >>> plt.show()\n\n ' ... def hypergeometric(self, ngood, nbad, nsample, size=...) -> typing.Any: ' Draw samples from a Hypergeometric distribution.\n\n Samples are drawn from a hypergeometric distribution with specified\n parameters, `ngood` (ways to make a good selection), `nbad` (ways to make\n a bad selection), and `nsample` (number of items sampled, which is less\n than or equal to the sum ``ngood + nbad``).\n\n Parameters\n ----------\n ngood : int or array_like of ints\n Number of ways to make a good selection. Must be nonnegative and\n less than 10**9.\n nbad : int or array_like of ints\n Number of ways to make a bad selection. Must be nonnegative and\n less than 10**9.\n nsample : int or array_like of ints\n Number of items sampled. Must be nonnegative and less than\n ``ngood + nbad``.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if `ngood`, `nbad`, and `nsample`\n are all scalars. Otherwise, ``np.broadcast(ngood, nbad, nsample).size``\n samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized hypergeometric distribution. Each\n sample is the number of good items within a randomly selected subset of\n size `nsample` taken from a set of `ngood` good items and `nbad` bad items.\n\n See Also\n --------\n multivariate_hypergeometric : Draw samples from the multivariate\n hypergeometric distribution.\n scipy.stats.hypergeom : probability density function, distribution or\n cumulative density function, etc.\n\n Notes\n -----\n The probability density for the Hypergeometric distribution is\n\n .. math:: P(x) = \\frac{\\binom{g}{x}\\binom{b}{n-x}}{\\binom{g+b}{n}},\n\n where :math:`0 \\le x \\le n` and :math:`n-b \\le x \\le g`\n\n for P(x) the probability of ``x`` good results in the drawn sample,\n g = `ngood`, b = `nbad`, and n = `nsample`.\n\n Consider an urn with black and white marbles in it, `ngood` of them\n are black and `nbad` are white. If you draw `nsample` balls without\n replacement, then the hypergeometric distribution describes the\n distribution of black balls in the drawn sample.\n\n Note that this distribution is very similar to the binomial\n distribution, except that in this case, samples are drawn without\n replacement, whereas in the Binomial case samples are drawn with\n replacement (or the sample space is infinite). As the sample space\n becomes large, this distribution approaches the binomial.\n\n The arguments `ngood` and `nbad` each must be less than `10**9`. For\n extremely large arguments, the algorithm that is used to compute the\n samples [4]_ breaks down because of loss of precision in floating point\n calculations. For such large values, if `nsample` is not also large,\n the distribution can be approximated with the binomial distribution,\n `binomial(n=nsample, p=ngood/(ngood + nbad))`.\n\n References\n ----------\n .. [1] Lentner, Marvin, "Elementary Applied Statistics", Bogden\n and Quigley, 1972.\n .. [2] Weisstein, Eric W. "Hypergeometric Distribution." From\n MathWorld--A Wolfram Web Resource.\n http://mathworld.wolfram.com/HypergeometricDistribution.html\n .. [3] Wikipedia, "Hypergeometric distribution",\n https://en.wikipedia.org/wiki/Hypergeometric_distribution\n .. [4] Stadlober, Ernst, "The ratio of uniforms approach for generating\n discrete random variates", Journal of Computational and Applied\n Mathematics, 31, pp. 181-189 (1990).\n\n Examples\n --------\n Draw samples from the distribution:\n\n >>> rng = np.random.default_rng()\n >>> ngood, nbad, nsamp = 100, 2, 10\n # number of good, number of bad, and number of samples\n >>> s = rng.hypergeometric(ngood, nbad, nsamp, 1000)\n >>> from matplotlib.pyplot import hist\n >>> hist(s)\n # note that it is very unlikely to grab both bad items\n\n Suppose you have an urn with 15 white and 15 black marbles.\n If you pull 15 marbles at random, how likely is it that\n 12 or more of them are one color?\n\n >>> s = rng.hypergeometric(15, 15, 15, 100000)\n >>> sum(s>=12)/100000. + sum(s<=3)/100000.\n # answer = 0.003 ... pretty unlikely!\n\n ' ... def integers(self, low, high=..., size=..., dtype=..., endpoint=...) -> typing.Any: ' Return random integers from `low` (inclusive) to `high` (exclusive), or\n if endpoint=True, `low` (inclusive) to `high` (inclusive). Replaces\n `RandomState.randint` (with endpoint=False) and\n `RandomState.random_integers` (with endpoint=True)\n\n Return random integers from the "discrete uniform" distribution of\n the specified dtype. If `high` is None (the default), then results are\n from 0 to `low`.\n\n Parameters\n ----------\n low : int or array-like of ints\n Lowest (signed) integers to be drawn from the distribution (unless\n ``high=None``, in which case this parameter is 0 and this value is\n used for `high`).\n high : int or array-like of ints, optional\n If provided, one above the largest (signed) integer to be drawn\n from the distribution (see above for behavior if ``high=None``).\n If array-like, must contain integer values\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. Default is None, in which case a\n single value is returned.\n dtype : dtype, optional\n Desired dtype of the result. Byteorder must be native.\n The default value is np.int64.\n endpoint : bool, optional\n If true, sample from the interval [low, high] instead of the\n default [low, high)\n Defaults to False\n\n Returns\n -------\n out : int or ndarray of ints\n `size`-shaped array of random integers from the appropriate\n distribution, or a single such random int if `size` not provided.\n\n Notes\n -----\n When using broadcasting with uint64 dtypes, the maximum value (2**64)\n cannot be represented as a standard integer type. The high array (or\n low if high is None) must have object dtype, e.g., array([2**64]).\n\n Examples\n --------\n >>> rng = np.random.default_rng()\n >>> rng.integers(2, size=10)\n array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) # random\n >>> rng.integers(1, size=10)\n array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])\n\n Generate a 2 x 4 array of ints between 0 and 4, inclusive:\n\n >>> rng.integers(5, size=(2, 4))\n array([[4, 0, 2, 1],\n [3, 2, 2, 0]]) # random\n\n Generate a 1 x 3 array with 3 different upper bounds\n\n >>> rng.integers(1, [3, 5, 10])\n array([2, 2, 9]) # random\n\n Generate a 1 by 3 array with 3 different lower bounds\n\n >>> rng.integers([1, 5, 7], 10)\n array([9, 8, 7]) # random\n\n Generate a 2 by 4 array using broadcasting with dtype of uint8\n\n >>> rng.integers([1, 3, 5, 7], [[10], [20]], dtype=np.uint8)\n array([[ 8, 6, 9, 7],\n [ 1, 16, 9, 12]], dtype=uint8) # random\n\n References\n ----------\n .. [1] Daniel Lemire., "Fast Random Integer Generation in an Interval",\n ACM Transactions on Modeling and Computer Simulation 29 (1), 2019,\n http://arxiv.org/abs/1805.10941.\n\n ' ... def laplace(self, loc=..., scale=..., size=...) -> typing.Any: ' Draw samples from the Laplace or double exponential distribution with\n specified location (or mean) and scale (decay).\n\n The Laplace distribution is similar to the Gaussian/normal distribution,\n but is sharper at the peak and has fatter tails. It represents the\n difference between two independent, identically distributed exponential\n random variables.\n\n Parameters\n ----------\n loc : float or array_like of floats, optional\n The position, :math:`\\mu`, of the distribution peak. Default is 0.\n scale : float or array_like of floats, optional\n :math:`\\lambda`, the exponential decay. Default is 1. Must be non-\n negative.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``loc`` and ``scale`` are both scalars.\n Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized Laplace distribution.\n\n Notes\n -----\n It has the probability density function\n\n .. math:: f(x; \\mu, \\lambda) = \\frac{1}{2\\lambda}\n \\exp\\left(-\\frac{|x - \\mu|}{\\lambda}\\right).\n\n The first law of Laplace, from 1774, states that the frequency\n of an error can be expressed as an exponential function of the\n absolute magnitude of the error, which leads to the Laplace\n distribution. For many problems in economics and health\n sciences, this distribution seems to model the data better\n than the standard Gaussian distribution.\n\n References\n ----------\n .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of\n Mathematical Functions with Formulas, Graphs, and Mathematical\n Tables, 9th printing," New York: Dover, 1972.\n .. [2] Kotz, Samuel, et. al. "The Laplace Distribution and\n Generalizations, " Birkhauser, 2001.\n .. [3] Weisstein, Eric W. "Laplace Distribution."\n From MathWorld--A Wolfram Web Resource.\n http://mathworld.wolfram.com/LaplaceDistribution.html\n .. [4] Wikipedia, "Laplace distribution",\n https://en.wikipedia.org/wiki/Laplace_distribution\n\n Examples\n --------\n Draw samples from the distribution\n\n >>> loc, scale = 0., 1.\n >>> s = np.random.default_rng().laplace(loc, scale, 1000)\n\n Display the histogram of the samples, along with\n the probability density function:\n\n >>> import matplotlib.pyplot as plt\n >>> count, bins, ignored = plt.hist(s, 30, density=True)\n >>> x = np.arange(-8., 8., .01)\n >>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale)\n >>> plt.plot(x, pdf)\n\n Plot Gaussian for comparison:\n\n >>> g = (1/(scale * np.sqrt(2 * np.pi)) *\n ... np.exp(-(x - loc)**2 / (2 * scale**2)))\n >>> plt.plot(x,g)\n\n ' ... def logistic(self, loc=..., scale=..., size=...) -> typing.Any: ' Draw samples from a logistic distribution.\n\n Samples are drawn from a logistic distribution with specified\n parameters, loc (location or mean, also median), and scale (>0).\n\n Parameters\n ----------\n loc : float or array_like of floats, optional\n Parameter of the distribution. Default is 0.\n scale : float or array_like of floats, optional\n Parameter of the distribution. Must be non-negative.\n Default is 1.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``loc`` and ``scale`` are both scalars.\n Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized logistic distribution.\n\n See Also\n --------\n scipy.stats.logistic : probability density function, distribution or\n cumulative density function, etc.\n\n Notes\n -----\n The probability density for the Logistic distribution is\n\n .. math:: P(x) = P(x) = \\frac{e^{-(x-\\mu)/s}}{s(1+e^{-(x-\\mu)/s})^2},\n\n where :math:`\\mu` = location and :math:`s` = scale.\n\n The Logistic distribution is used in Extreme Value problems where it\n can act as a mixture of Gumbel distributions, in Epidemiology, and by\n the World Chess Federation (FIDE) where it is used in the Elo ranking\n system, assuming the performance of each player is a logistically\n distributed random variable.\n\n References\n ----------\n .. [1] Reiss, R.-D. and Thomas M. (2001), "Statistical Analysis of\n Extreme Values, from Insurance, Finance, Hydrology and Other\n Fields," Birkhauser Verlag, Basel, pp 132-133.\n .. [2] Weisstein, Eric W. "Logistic Distribution." From\n MathWorld--A Wolfram Web Resource.\n http://mathworld.wolfram.com/LogisticDistribution.html\n .. [3] Wikipedia, "Logistic-distribution",\n https://en.wikipedia.org/wiki/Logistic_distribution\n\n Examples\n --------\n Draw samples from the distribution:\n\n >>> loc, scale = 10, 1\n >>> s = np.random.default_rng().logistic(loc, scale, 10000)\n >>> import matplotlib.pyplot as plt\n >>> count, bins, ignored = plt.hist(s, bins=50)\n\n # plot against distribution\n\n >>> def logist(x, loc, scale):\n ... return np.exp((loc-x)/scale)/(scale*(1+np.exp((loc-x)/scale))**2)\n >>> lgst_val = logist(bins, loc, scale)\n >>> plt.plot(bins, lgst_val * count.max() / lgst_val.max())\n >>> plt.show()\n\n ' ... def lognormal(self, mean=..., sigma=..., size=...) -> typing.Any: ' Draw samples from a log-normal distribution.\n\n Draw samples from a log-normal distribution with specified mean,\n standard deviation, and array shape. Note that the mean and standard\n deviation are not the values for the distribution itself, but of the\n underlying normal distribution it is derived from.\n\n Parameters\n ----------\n mean : float or array_like of floats, optional\n Mean value of the underlying normal distribution. Default is 0.\n sigma : float or array_like of floats, optional\n Standard deviation of the underlying normal distribution. Must be\n non-negative. Default is 1.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``mean`` and ``sigma`` are both scalars.\n Otherwise, ``np.broadcast(mean, sigma).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized log-normal distribution.\n\n See Also\n --------\n scipy.stats.lognorm : probability density function, distribution,\n cumulative density function, etc.\n\n Notes\n -----\n A variable `x` has a log-normal distribution if `log(x)` is normally\n distributed. The probability density function for the log-normal\n distribution is:\n\n .. math:: p(x) = \\frac{1}{\\sigma x \\sqrt{2\\pi}}\n e^{(-\\frac{(ln(x)-\\mu)^2}{2\\sigma^2})}\n\n where :math:`\\mu` is the mean and :math:`\\sigma` is the standard\n deviation of the normally distributed logarithm of the variable.\n A log-normal distribution results if a random variable is the *product*\n of a large number of independent, identically-distributed variables in\n the same way that a normal distribution results if the variable is the\n *sum* of a large number of independent, identically-distributed\n variables.\n\n References\n ----------\n .. [1] Limpert, E., Stahel, W. A., and Abbt, M., "Log-normal\n Distributions across the Sciences: Keys and Clues,"\n BioScience, Vol. 51, No. 5, May, 2001.\n https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf\n .. [2] Reiss, R.D. and Thomas, M., "Statistical Analysis of Extreme\n Values," Basel: Birkhauser Verlag, 2001, pp. 31-32.\n\n Examples\n --------\n Draw samples from the distribution:\n\n >>> rng = np.random.default_rng()\n >>> mu, sigma = 3., 1. # mean and standard deviation\n >>> s = rng.lognormal(mu, sigma, 1000)\n\n Display the histogram of the samples, along with\n the probability density function:\n\n >>> import matplotlib.pyplot as plt\n >>> count, bins, ignored = plt.hist(s, 100, density=True, align=\'mid\')\n\n >>> x = np.linspace(min(bins), max(bins), 10000)\n >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))\n ... / (x * sigma * np.sqrt(2 * np.pi)))\n\n >>> plt.plot(x, pdf, linewidth=2, color=\'r\')\n >>> plt.axis(\'tight\')\n >>> plt.show()\n\n Demonstrate that taking the products of random samples from a uniform\n distribution can be fit well by a log-normal probability density\n function.\n\n >>> # Generate a thousand samples: each is the product of 100 random\n >>> # values, drawn from a normal distribution.\n >>> rng = rng\n >>> b = []\n >>> for i in range(1000):\n ... a = 10. + rng.standard_normal(100)\n ... b.append(np.prod(a))\n\n >>> b = np.array(b) / np.min(b) # scale values to be positive\n >>> count, bins, ignored = plt.hist(b, 100, density=True, align=\'mid\')\n >>> sigma = np.std(np.log(b))\n >>> mu = np.mean(np.log(b))\n\n >>> x = np.linspace(min(bins), max(bins), 10000)\n >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))\n ... / (x * sigma * np.sqrt(2 * np.pi)))\n\n >>> plt.plot(x, pdf, color=\'r\', linewidth=2)\n >>> plt.show()\n\n ' ... def logseries(self, p, size=...) -> typing.Any: ' Draw samples from a logarithmic series distribution.\n\n Samples are drawn from a log series distribution with specified\n shape parameter, 0 <= ``p`` < 1.\n\n Parameters\n ----------\n p : float or array_like of floats\n Shape parameter for the distribution. Must be in the range [0, 1).\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``p`` is a scalar. Otherwise,\n ``np.array(p).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized logarithmic series distribution.\n\n See Also\n --------\n scipy.stats.logser : probability density function, distribution or\n cumulative density function, etc.\n\n Notes\n -----\n The probability mass function for the Log Series distribution is\n\n .. math:: P(k) = \\frac{-p^k}{k \\ln(1-p)},\n\n where p = probability.\n\n The log series distribution is frequently used to represent species\n richness and occurrence, first proposed by Fisher, Corbet, and\n Williams in 1943 [2]. It may also be used to model the numbers of\n occupants seen in cars [3].\n\n References\n ----------\n .. [1] Buzas, Martin A.; Culver, Stephen J., Understanding regional\n species diversity through the log series distribution of\n occurrences: BIODIVERSITY RESEARCH Diversity & Distributions,\n Volume 5, Number 5, September 1999 , pp. 187-195(9).\n .. [2] Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The\n relation between the number of species and the number of\n individuals in a random sample of an animal population.\n Journal of Animal Ecology, 12:42-58.\n .. [3] D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small\n Data Sets, CRC Press, 1994.\n .. [4] Wikipedia, "Logarithmic distribution",\n https://en.wikipedia.org/wiki/Logarithmic_distribution\n\n Examples\n --------\n Draw samples from the distribution:\n\n >>> a = .6\n >>> s = np.random.default_rng().logseries(a, 10000)\n >>> import matplotlib.pyplot as plt\n >>> count, bins, ignored = plt.hist(s)\n\n # plot against distribution\n\n >>> def logseries(k, p):\n ... return -p**k/(k*np.log(1-p))\n >>> plt.plot(bins, logseries(bins, a) * count.max()/\n ... logseries(bins, a).max(), \'r\')\n >>> plt.show()\n\n ' ... def multinomial(self, n, pvals, size=...) -> typing.Any: ' Draw samples from a multinomial distribution.\n\n The multinomial distribution is a multivariate generalization of the\n binomial distribution. Take an experiment with one of ``p``\n possible outcomes. An example of such an experiment is throwing a dice,\n where the outcome can be 1 through 6. Each sample drawn from the\n distribution represents `n` such experiments. Its values,\n ``X_i = [X_0, X_1, ..., X_p]``, represent the number of times the\n outcome was ``i``.\n\n Parameters\n ----------\n n : int or array-like of ints\n Number of experiments.\n pvals : array-like of floats\n Probabilities of each of the ``p`` different outcomes with shape\n ``(k0, k1, ..., kn, p)``. Each element ``pvals[i,j,...,:]`` must\n sum to 1 (however, the last element is always assumed to account\n for the remaining probability, as long as\n ``sum(pvals[..., :-1], axis=-1) <= 1.0``. Must have at least 1\n dimension where pvals.shape[-1] > 0.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn each with ``p`` elements. Default\n is None where the output size is determined by the broadcast shape\n of ``n`` and all by the final dimension of ``pvals``, which is\n denoted as ``b=(b0, b1, ..., bq)``. If size is not None, then it\n must be compatible with the broadcast shape ``b``. Specifically,\n size must have ``q`` or more elements and size[-(q-j):] must equal\n ``bj``.\n\n Returns\n -------\n out : ndarray\n The drawn samples, of shape size, if provided. When size is\n provided, the output shape is size + (p,) If not specified,\n the shape is determined by the broadcast shape of ``n`` and\n ``pvals``, ``(b0, b1, ..., bq)`` augmented with the dimension of\n the multinomial, ``p``, so that that output shape is\n ``(b0, b1, ..., bq, p)``.\n\n Each entry ``out[i,j,...,:]`` is a ``p``-dimensional value drawn\n from the distribution.\n\n .. versionchanged:: 1.22.0\n Added support for broadcasting `pvals` against `n`\n\n Examples\n --------\n Throw a dice 20 times:\n\n >>> rng = np.random.default_rng()\n >>> rng.multinomial(20, [1/6.]*6, size=1)\n array([[4, 1, 7, 5, 2, 1]]) # random\n\n It landed 4 times on 1, once on 2, etc.\n\n Now, throw the dice 20 times, and 20 times again:\n\n >>> rng.multinomial(20, [1/6.]*6, size=2)\n array([[3, 4, 3, 3, 4, 3],\n [2, 4, 3, 4, 0, 7]]) # random\n\n For the first run, we threw 3 times 1, 4 times 2, etc. For the second,\n we threw 2 times 1, 4 times 2, etc.\n\n Now, do one experiment throwing the dice 10 time, and 10 times again,\n and another throwing the dice 20 times, and 20 times again:\n\n >>> rng.multinomial([[10], [20]], [1/6.]*6, size=(2, 2))\n array([[[2, 4, 0, 1, 2, 1],\n [1, 3, 0, 3, 1, 2]],\n [[1, 4, 4, 4, 4, 3],\n [3, 3, 2, 5, 5, 2]]]) # random\n\n The first array shows the outcomes of throwing the dice 10 times, and\n the second shows the outcomes from throwing the dice 20 times.\n\n A loaded die is more likely to land on number 6:\n\n >>> rng.multinomial(100, [1/7.]*5 + [2/7.])\n array([11, 16, 14, 17, 16, 26]) # random\n\n Simulate 10 throws of a 4-sided die and 20 throws of a 6-sided die\n\n >>> rng.multinomial([10, 20],[[1/4]*4 + [0]*2, [1/6]*6])\n array([[2, 1, 4, 3, 0, 0],\n [3, 3, 3, 6, 1, 4]], dtype=int64) # random\n\n Generate categorical random variates from two categories where the\n first has 3 outcomes and the second has 2.\n\n >>> rng.multinomial(1, [[.1, .5, .4 ], [.3, .7, .0]])\n array([[0, 0, 1],\n [0, 1, 0]], dtype=int64) # random\n\n ``argmax(axis=-1)`` is then used to return the categories.\n\n >>> pvals = [[.1, .5, .4 ], [.3, .7, .0]]\n >>> rvs = rng.multinomial(1, pvals, size=(4,2))\n >>> rvs.argmax(axis=-1)\n array([[0, 1],\n [2, 0],\n [2, 1],\n [2, 0]], dtype=int64) # random\n\n The same output dimension can be produced using broadcasting.\n\n >>> rvs = rng.multinomial([[1]] * 4, pvals)\n >>> rvs.argmax(axis=-1)\n array([[0, 1],\n [2, 0],\n [2, 1],\n [2, 0]], dtype=int64) # random\n\n The probability inputs should be normalized. As an implementation\n detail, the value of the last entry is ignored and assumed to take\n up any leftover probability mass, but this should not be relied on.\n A biased coin which has twice as much weight on one side as on the\n other should be sampled like so:\n\n >>> rng.multinomial(100, [1.0 / 3, 2.0 / 3]) # RIGHT\n array([38, 62]) # random\n\n not like:\n\n >>> rng.multinomial(100, [1.0, 2.0]) # WRONG\n Traceback (most recent call last):\n ValueError: pvals < 0, pvals > 1 or pvals contains NaNs\n ' ... def multivariate_hypergeometric(self, colors, nsample, size=..., method=...) -> typing.Any: ' Generate variates from a multivariate hypergeometric distribution.\n\n The multivariate hypergeometric distribution is a generalization\n of the hypergeometric distribution.\n\n Choose ``nsample`` items at random without replacement from a\n collection with ``N`` distinct types. ``N`` is the length of\n ``colors``, and the values in ``colors`` are the number of occurrences\n of that type in the collection. The total number of items in the\n collection is ``sum(colors)``. Each random variate generated by this\n function is a vector of length ``N`` holding the counts of the\n different types that occurred in the ``nsample`` items.\n\n The name ``colors`` comes from a common description of the\n distribution: it is the probability distribution of the number of\n marbles of each color selected without replacement from an urn\n containing marbles of different colors; ``colors[i]`` is the number\n of marbles in the urn with color ``i``.\n\n Parameters\n ----------\n colors : sequence of integers\n The number of each type of item in the collection from which\n a sample is drawn. The values in ``colors`` must be nonnegative.\n To avoid loss of precision in the algorithm, ``sum(colors)``\n must be less than ``10**9`` when `method` is "marginals".\n nsample : int\n The number of items selected. ``nsample`` must not be greater\n than ``sum(colors)``.\n size : int or tuple of ints, optional\n The number of variates to generate, either an integer or a tuple\n holding the shape of the array of variates. If the given size is,\n e.g., ``(k, m)``, then ``k * m`` variates are drawn, where one\n variate is a vector of length ``len(colors)``, and the return value\n has shape ``(k, m, len(colors))``. If `size` is an integer, the\n output has shape ``(size, len(colors))``. Default is None, in\n which case a single variate is returned as an array with shape\n ``(len(colors),)``.\n method : string, optional\n Specify the algorithm that is used to generate the variates.\n Must be \'count\' or \'marginals\' (the default). See the Notes\n for a description of the methods.\n\n Returns\n -------\n variates : ndarray\n Array of variates drawn from the multivariate hypergeometric\n distribution.\n\n See Also\n --------\n hypergeometric : Draw samples from the (univariate) hypergeometric\n distribution.\n\n Notes\n -----\n The two methods do not return the same sequence of variates.\n\n The "count" algorithm is roughly equivalent to the following numpy\n code::\n\n choices = np.repeat(np.arange(len(colors)), colors)\n selection = np.random.choice(choices, nsample, replace=False)\n variate = np.bincount(selection, minlength=len(colors))\n\n The "count" algorithm uses a temporary array of integers with length\n ``sum(colors)``.\n\n The "marginals" algorithm generates a variate by using repeated\n calls to the univariate hypergeometric sampler. It is roughly\n equivalent to::\n\n variate = np.zeros(len(colors), dtype=np.int64)\n # `remaining` is the cumulative sum of `colors` from the last\n # element to the first; e.g. if `colors` is [3, 1, 5], then\n # `remaining` is [9, 6, 5].\n remaining = np.cumsum(colors[::-1])[::-1]\n for i in range(len(colors)-1):\n if nsample < 1:\n break\n variate[i] = hypergeometric(colors[i], remaining[i+1],\n nsample)\n nsample -= variate[i]\n variate[-1] = nsample\n\n The default method is "marginals". For some cases (e.g. when\n `colors` contains relatively small integers), the "count" method\n can be significantly faster than the "marginals" method. If\n performance of the algorithm is important, test the two methods\n with typical inputs to decide which works best.\n\n .. versionadded:: 1.18.0\n\n Examples\n --------\n >>> colors = [16, 8, 4]\n >>> seed = 4861946401452\n >>> gen = np.random.Generator(np.random.PCG64(seed))\n >>> gen.multivariate_hypergeometric(colors, 6)\n array([5, 0, 1])\n >>> gen.multivariate_hypergeometric(colors, 6, size=3)\n array([[5, 0, 1],\n [2, 2, 2],\n [3, 3, 0]])\n >>> gen.multivariate_hypergeometric(colors, 6, size=(2, 2))\n array([[[3, 2, 1],\n [3, 2, 1]],\n [[4, 1, 1],\n [3, 2, 1]]])\n ' ... def multivariate_normal(self, mean, cov, size=..., check_valid=..., tol=..., *, method=...) -> typing.Any: ' Draw random samples from a multivariate normal distribution.\n\n The multivariate normal, multinormal or Gaussian distribution is a\n generalization of the one-dimensional normal distribution to higher\n dimensions. Such a distribution is specified by its mean and\n covariance matrix. These parameters are analogous to the mean\n (average or "center") and variance (the squared standard deviation,\n or "width") of the one-dimensional normal distribution.\n\n Parameters\n ----------\n mean : 1-D array_like, of length N\n Mean of the N-dimensional distribution.\n cov : 2-D array_like, of shape (N, N)\n Covariance matrix of the distribution. It must be symmetric and\n positive-semidefinite for proper sampling.\n size : int or tuple of ints, optional\n Given a shape of, for example, ``(m,n,k)``, ``m*n*k`` samples are\n generated, and packed in an `m`-by-`n`-by-`k` arrangement. Because\n each sample is `N`-dimensional, the output shape is ``(m,n,k,N)``.\n If no shape is specified, a single (`N`-D) sample is returned.\n check_valid : { \'warn\', \'raise\', \'ignore\' }, optional\n Behavior when the covariance matrix is not positive semidefinite.\n tol : float, optional\n Tolerance when checking the singular values in covariance matrix.\n cov is cast to double before the check.\n method : { \'svd\', \'eigh\', \'cholesky\'}, optional\n The cov input is used to compute a factor matrix A such that\n ``A @ A.T = cov``. This argument is used to select the method\n used to compute the factor matrix A. The default method \'svd\' is\n the slowest, while \'cholesky\' is the fastest but less robust than\n the slowest method. The method `eigh` uses eigen decomposition to\n compute A and is faster than svd but slower than cholesky.\n\n .. versionadded:: 1.18.0\n\n Returns\n -------\n out : ndarray\n The drawn samples, of shape *size*, if that was provided. If not,\n the shape is ``(N,)``.\n\n In other words, each entry ``out[i,j,...,:]`` is an N-dimensional\n value drawn from the distribution.\n\n Notes\n -----\n The mean is a coordinate in N-dimensional space, which represents the\n location where samples are most likely to be generated. This is\n analogous to the peak of the bell curve for the one-dimensional or\n univariate normal distribution.\n\n Covariance indicates the level to which two variables vary together.\n From the multivariate normal distribution, we draw N-dimensional\n samples, :math:`X = [x_1, x_2, ... x_N]`. The covariance matrix\n element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`.\n The element :math:`C_{ii}` is the variance of :math:`x_i` (i.e. its\n "spread").\n\n Instead of specifying the full covariance matrix, popular\n approximations include:\n\n - Spherical covariance (`cov` is a multiple of the identity matrix)\n - Diagonal covariance (`cov` has non-negative elements, and only on\n the diagonal)\n\n This geometrical property can be seen in two dimensions by plotting\n generated data-points:\n\n >>> mean = [0, 0]\n >>> cov = [[1, 0], [0, 100]] # diagonal covariance\n\n Diagonal covariance means that points are oriented along x or y-axis:\n\n >>> import matplotlib.pyplot as plt\n >>> x, y = np.random.default_rng().multivariate_normal(mean, cov, 5000).T\n >>> plt.plot(x, y, \'x\')\n >>> plt.axis(\'equal\')\n >>> plt.show()\n\n Note that the covariance matrix must be positive semidefinite (a.k.a.\n nonnegative-definite). Otherwise, the behavior of this method is\n undefined and backwards compatibility is not guaranteed.\n\n This function internally uses linear algebra routines, and thus results\n may not be identical (even up to precision) across architectures, OSes,\n or even builds. For example, this is likely if ``cov`` has multiple equal\n singular values and ``method`` is ``\'svd\'`` (default). In this case,\n ``method=\'cholesky\'`` may be more robust.\n\n References\n ----------\n .. [1] Papoulis, A., "Probability, Random Variables, and Stochastic\n Processes," 3rd ed., New York: McGraw-Hill, 1991.\n .. [2] Duda, R. O., Hart, P. E., and Stork, D. G., "Pattern\n Classification," 2nd ed., New York: Wiley, 2001.\n\n Examples\n --------\n >>> mean = (1, 2)\n >>> cov = [[1, 0], [0, 1]]\n >>> rng = np.random.default_rng()\n >>> x = rng.multivariate_normal(mean, cov, (3, 3))\n >>> x.shape\n (3, 3, 2)\n\n We can use a different method other than the default to factorize cov:\n\n >>> y = rng.multivariate_normal(mean, cov, (3, 3), method=\'cholesky\')\n >>> y.shape\n (3, 3, 2)\n\n Here we generate 800 samples from the bivariate normal distribution\n with mean [0, 0] and covariance matrix [[6, -3], [-3, 3.5]]. The\n expected variances of the first and second components of the sample\n are 6 and 3.5, respectively, and the expected correlation\n coefficient is -3/sqrt(6*3.5) ≈ -0.65465.\n\n >>> cov = np.array([[6, -3], [-3, 3.5]])\n >>> pts = rng.multivariate_normal([0, 0], cov, size=800)\n\n Check that the mean, covariance, and correlation coefficient of the\n sample are close to the expected values:\n\n >>> pts.mean(axis=0)\n array([ 0.0326911 , -0.01280782]) # may vary\n >>> np.cov(pts.T)\n array([[ 5.96202397, -2.85602287],\n [-2.85602287, 3.47613949]]) # may vary\n >>> np.corrcoef(pts.T)[0, 1]\n -0.6273591314603949 # may vary\n\n We can visualize this data with a scatter plot. The orientation\n of the point cloud illustrates the negative correlation of the\n components of this sample.\n\n >>> import matplotlib.pyplot as plt\n >>> plt.plot(pts[:, 0], pts[:, 1], \'.\', alpha=0.5)\n >>> plt.axis(\'equal\')\n >>> plt.grid()\n >>> plt.show()\n ' ... def negative_binomial(self, n, p, size=...) -> typing.Any: ' Draw samples from a negative binomial distribution.\n\n Samples are drawn from a negative binomial distribution with specified\n parameters, `n` successes and `p` probability of success where `n`\n is > 0 and `p` is in the interval (0, 1].\n\n Parameters\n ----------\n n : float or array_like of floats\n Parameter of the distribution, > 0.\n p : float or array_like of floats\n Parameter of the distribution. Must satisfy 0 < p <= 1.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``n`` and ``p`` are both scalars.\n Otherwise, ``np.broadcast(n, p).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized negative binomial distribution,\n where each sample is equal to N, the number of failures that\n occurred before a total of n successes was reached.\n\n Notes\n -----\n The probability mass function of the negative binomial distribution is\n\n .. math:: P(N;n,p) = \\frac{\\Gamma(N+n)}{N!\\Gamma(n)}p^{n}(1-p)^{N},\n\n where :math:`n` is the number of successes, :math:`p` is the\n probability of success, :math:`N+n` is the number of trials, and\n :math:`\\Gamma` is the gamma function. When :math:`n` is an integer,\n :math:`\\frac{\\Gamma(N+n)}{N!\\Gamma(n)} = \\binom{N+n-1}{N}`, which is\n the more common form of this term in the pmf. The negative\n binomial distribution gives the probability of N failures given n\n successes, with a success on the last trial.\n\n If one throws a die repeatedly until the third time a "1" appears,\n then the probability distribution of the number of non-"1"s that\n appear before the third "1" is a negative binomial distribution.\n\n Because this method internally calls ``Generator.poisson`` with an\n intermediate random value, a ValueError is raised when the choice of \n :math:`n` and :math:`p` would result in the mean + 10 sigma of the sampled\n intermediate distribution exceeding the max acceptable value of the \n ``Generator.poisson`` method. This happens when :math:`p` is too low \n (a lot of failures happen for every success) and :math:`n` is too big (\n a lot of successes are allowed).\n Therefore, the :math:`n` and :math:`p` values must satisfy the constraint:\n\n .. math:: n\\frac{1-p}{p}+10n\\sqrt{n}\\frac{1-p}{p}<2^{63}-1-10\\sqrt{2^{63}-1},\n\n Where the left side of the equation is the derived mean + 10 sigma of\n a sample from the gamma distribution internally used as the :math:`lam`\n parameter of a poisson sample, and the right side of the equation is\n the constraint for maximum value of :math:`lam` in ``Generator.poisson``.\n\n References\n ----------\n .. [1] Weisstein, Eric W. "Negative Binomial Distribution." From\n MathWorld--A Wolfram Web Resource.\n http://mathworld.wolfram.com/NegativeBinomialDistribution.html\n .. [2] Wikipedia, "Negative binomial distribution",\n https://en.wikipedia.org/wiki/Negative_binomial_distribution\n\n Examples\n --------\n Draw samples from the distribution:\n\n A real world example. A company drills wild-cat oil\n exploration wells, each with an estimated probability of\n success of 0.1. What is the probability of having one success\n for each successive well, that is what is the probability of a\n single success after drilling 5 wells, after 6 wells, etc.?\n\n >>> s = np.random.default_rng().negative_binomial(1, 0.1, 100000)\n >>> for i in range(1, 11): # doctest: +SKIP\n ... probability = sum(s>> rng = np.random.default_rng()\n >>> import matplotlib.pyplot as plt\n >>> values = plt.hist(rng.noncentral_chisquare(3, 20, 100000),\n ... bins=200, density=True)\n >>> plt.show()\n\n Draw values from a noncentral chisquare with very small noncentrality,\n and compare to a chisquare.\n\n >>> plt.figure()\n >>> values = plt.hist(rng.noncentral_chisquare(3, .0000001, 100000),\n ... bins=np.arange(0., 25, .1), density=True)\n >>> values2 = plt.hist(rng.chisquare(3, 100000),\n ... bins=np.arange(0., 25, .1), density=True)\n >>> plt.plot(values[1][0:-1], values[0]-values2[0], \'ob\')\n >>> plt.show()\n\n Demonstrate how large values of non-centrality lead to a more symmetric\n distribution.\n\n >>> plt.figure()\n >>> values = plt.hist(rng.noncentral_chisquare(3, 20, 100000),\n ... bins=200, density=True)\n >>> plt.show()\n\n ' ... def noncentral_f(self, dfnum, dfden, nonc, size=...) -> typing.Any: ' Draw samples from the noncentral F distribution.\n\n Samples are drawn from an F distribution with specified parameters,\n `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of\n freedom in denominator), where both parameters > 1.\n `nonc` is the non-centrality parameter.\n\n Parameters\n ----------\n dfnum : float or array_like of floats\n Numerator degrees of freedom, must be > 0.\n\n .. versionchanged:: 1.14.0\n Earlier NumPy versions required dfnum > 1.\n dfden : float or array_like of floats\n Denominator degrees of freedom, must be > 0.\n nonc : float or array_like of floats\n Non-centrality parameter, the sum of the squares of the numerator\n means, must be >= 0.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``dfnum``, ``dfden``, and ``nonc``\n are all scalars. Otherwise, ``np.broadcast(dfnum, dfden, nonc).size``\n samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized noncentral Fisher distribution.\n\n Notes\n -----\n When calculating the power of an experiment (power = probability of\n rejecting the null hypothesis when a specific alternative is true) the\n non-central F statistic becomes important. When the null hypothesis is\n true, the F statistic follows a central F distribution. When the null\n hypothesis is not true, then it follows a non-central F statistic.\n\n References\n ----------\n .. [1] Weisstein, Eric W. "Noncentral F-Distribution."\n From MathWorld--A Wolfram Web Resource.\n http://mathworld.wolfram.com/NoncentralF-Distribution.html\n .. [2] Wikipedia, "Noncentral F-distribution",\n https://en.wikipedia.org/wiki/Noncentral_F-distribution\n\n Examples\n --------\n In a study, testing for a specific alternative to the null hypothesis\n requires use of the Noncentral F distribution. We need to calculate the\n area in the tail of the distribution that exceeds the value of the F\n distribution for the null hypothesis. We\'ll plot the two probability\n distributions for comparison.\n\n >>> rng = np.random.default_rng()\n >>> dfnum = 3 # between group deg of freedom\n >>> dfden = 20 # within groups degrees of freedom\n >>> nonc = 3.0\n >>> nc_vals = rng.noncentral_f(dfnum, dfden, nonc, 1000000)\n >>> NF = np.histogram(nc_vals, bins=50, density=True)\n >>> c_vals = rng.f(dfnum, dfden, 1000000)\n >>> F = np.histogram(c_vals, bins=50, density=True)\n >>> import matplotlib.pyplot as plt\n >>> plt.plot(F[1][1:], F[0])\n >>> plt.plot(NF[1][1:], NF[0])\n >>> plt.show()\n\n ' ... def normal(self, loc=..., scale=..., size=...) -> typing.Any: ' Draw random samples from a normal (Gaussian) distribution.\n\n The probability density function of the normal distribution, first\n derived by De Moivre and 200 years later by both Gauss and Laplace\n independently [2]_, is often called the bell curve because of\n its characteristic shape (see the example below).\n\n The normal distributions occurs often in nature. For example, it\n describes the commonly occurring distribution of samples influenced\n by a large number of tiny, random disturbances, each with its own\n unique distribution [2]_.\n\n Parameters\n ----------\n loc : float or array_like of floats\n Mean ("centre") of the distribution.\n scale : float or array_like of floats\n Standard deviation (spread or "width") of the distribution. Must be\n non-negative.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``loc`` and ``scale`` are both scalars.\n Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized normal distribution.\n\n See Also\n --------\n scipy.stats.norm : probability density function, distribution or\n cumulative density function, etc.\n\n Notes\n -----\n The probability density for the Gaussian distribution is\n\n .. math:: p(x) = \\frac{1}{\\sqrt{ 2 \\pi \\sigma^2 }}\n e^{ - \\frac{ (x - \\mu)^2 } {2 \\sigma^2} },\n\n where :math:`\\mu` is the mean and :math:`\\sigma` the standard\n deviation. The square of the standard deviation, :math:`\\sigma^2`,\n is called the variance.\n\n The function has its peak at the mean, and its "spread" increases with\n the standard deviation (the function reaches 0.607 times its maximum at\n :math:`x + \\sigma` and :math:`x - \\sigma` [2]_). This implies that\n :meth:`normal` is more likely to return samples lying close to the\n mean, rather than those far away.\n\n References\n ----------\n .. [1] Wikipedia, "Normal distribution",\n https://en.wikipedia.org/wiki/Normal_distribution\n .. [2] P. R. Peebles Jr., "Central Limit Theorem" in "Probability,\n Random Variables and Random Signal Principles", 4th ed., 2001,\n pp. 51, 51, 125.\n\n Examples\n --------\n Draw samples from the distribution:\n\n >>> mu, sigma = 0, 0.1 # mean and standard deviation\n >>> s = np.random.default_rng().normal(mu, sigma, 1000)\n\n Verify the mean and the variance:\n\n >>> abs(mu - np.mean(s))\n 0.0 # may vary\n\n >>> abs(sigma - np.std(s, ddof=1))\n 0.0 # may vary\n\n Display the histogram of the samples, along with\n the probability density function:\n\n >>> import matplotlib.pyplot as plt\n >>> count, bins, ignored = plt.hist(s, 30, density=True)\n >>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *\n ... np.exp( - (bins - mu)**2 / (2 * sigma**2) ),\n ... linewidth=2, color=\'r\')\n >>> plt.show()\n\n Two-by-four array of samples from the normal distribution with\n mean 3 and standard deviation 2.5:\n\n >>> np.random.default_rng().normal(3, 2.5, size=(2, 4))\n array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random\n [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random\n\n ' ... def pareto(self, a, size=...) -> typing.Any: ' Draw samples from a Pareto II or Lomax distribution with\n specified shape.\n\n The Lomax or Pareto II distribution is a shifted Pareto\n distribution. The classical Pareto distribution can be\n obtained from the Lomax distribution by adding 1 and\n multiplying by the scale parameter ``m`` (see Notes). The\n smallest value of the Lomax distribution is zero while for the\n classical Pareto distribution it is ``mu``, where the standard\n Pareto distribution has location ``mu = 1``. Lomax can also\n be considered as a simplified version of the Generalized\n Pareto distribution (available in SciPy), with the scale set\n to one and the location set to zero.\n\n The Pareto distribution must be greater than zero, and is\n unbounded above. It is also known as the "80-20 rule". In\n this distribution, 80 percent of the weights are in the lowest\n 20 percent of the range, while the other 20 percent fill the\n remaining 80 percent of the range.\n\n Parameters\n ----------\n a : float or array_like of floats\n Shape of the distribution. Must be positive.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``a`` is a scalar. Otherwise,\n ``np.array(a).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized Pareto distribution.\n\n See Also\n --------\n scipy.stats.lomax : probability density function, distribution or\n cumulative density function, etc.\n scipy.stats.genpareto : probability density function, distribution or\n cumulative density function, etc.\n\n Notes\n -----\n The probability density for the Pareto distribution is\n\n .. math:: p(x) = \\frac{am^a}{x^{a+1}}\n\n where :math:`a` is the shape and :math:`m` the scale.\n\n The Pareto distribution, named after the Italian economist\n Vilfredo Pareto, is a power law probability distribution\n useful in many real world problems. Outside the field of\n economics it is generally referred to as the Bradford\n distribution. Pareto developed the distribution to describe\n the distribution of wealth in an economy. It has also found\n use in insurance, web page access statistics, oil field sizes,\n and many other problems, including the download frequency for\n projects in Sourceforge [1]_. It is one of the so-called\n "fat-tailed" distributions.\n\n\n References\n ----------\n .. [1] Francis Hunt and Paul Johnson, On the Pareto Distribution of\n Sourceforge projects.\n .. [2] Pareto, V. (1896). Course of Political Economy. Lausanne.\n .. [3] Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme\n Values, Birkhauser Verlag, Basel, pp 23-30.\n .. [4] Wikipedia, "Pareto distribution",\n https://en.wikipedia.org/wiki/Pareto_distribution\n\n Examples\n --------\n Draw samples from the distribution:\n\n >>> a, m = 3., 2. # shape and mode\n >>> s = (np.random.default_rng().pareto(a, 1000) + 1) * m\n\n Display the histogram of the samples, along with the probability\n density function:\n\n >>> import matplotlib.pyplot as plt\n >>> count, bins, _ = plt.hist(s, 100, density=True)\n >>> fit = a*m**a / bins**(a+1)\n >>> plt.plot(bins, max(count)*fit/max(fit), linewidth=2, color=\'r\')\n >>> plt.show()\n\n ' ... def permutation(self, x, axis=...) -> typing.Any: ' Randomly permute a sequence, or return a permuted range.\n\n Parameters\n ----------\n x : int or array_like\n If `x` is an integer, randomly permute ``np.arange(x)``.\n If `x` is an array, make a copy and shuffle the elements\n randomly.\n axis : int, optional\n The axis which `x` is shuffled along. Default is 0.\n\n Returns\n -------\n out : ndarray\n Permuted sequence or array range.\n\n Examples\n --------\n >>> rng = np.random.default_rng()\n >>> rng.permutation(10)\n array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6]) # random\n\n >>> rng.permutation([1, 4, 9, 12, 15])\n array([15, 1, 9, 4, 12]) # random\n\n >>> arr = np.arange(9).reshape((3, 3))\n >>> rng.permutation(arr)\n array([[6, 7, 8], # random\n [0, 1, 2],\n [3, 4, 5]])\n\n >>> rng.permutation("abc")\n Traceback (most recent call last):\n ...\n numpy.exceptions.AxisError: axis 0 is out of bounds for array of dimension 0\n\n >>> arr = np.arange(9).reshape((3, 3))\n >>> rng.permutation(arr, axis=1)\n array([[0, 2, 1], # random\n [3, 5, 4],\n [6, 8, 7]])\n\n ' ... def permuted(self, x, axis=..., out=...) -> typing.Any: ' Randomly permute `x` along axis `axis`.\n\n Unlike `shuffle`, each slice along the given axis is shuffled\n independently of the others.\n\n Parameters\n ----------\n x : array_like, at least one-dimensional\n Array to be shuffled.\n axis : int, optional\n Slices of `x` in this axis are shuffled. Each slice\n is shuffled independently of the others. If `axis` is\n None, the flattened array is shuffled.\n out : ndarray, optional\n If given, this is the destination of the shuffled array.\n If `out` is None, a shuffled copy of the array is returned.\n\n Returns\n -------\n ndarray\n If `out` is None, a shuffled copy of `x` is returned.\n Otherwise, the shuffled array is stored in `out`,\n and `out` is returned\n\n See Also\n --------\n shuffle\n permutation\n \n Notes\n -----\n An important distinction between methods ``shuffle`` and ``permuted`` is \n how they both treat the ``axis`` parameter which can be found at \n :ref:`generator-handling-axis-parameter`.\n\n Examples\n --------\n Create a `numpy.random.Generator` instance:\n\n >>> rng = np.random.default_rng()\n\n Create a test array:\n\n >>> x = np.arange(24).reshape(3, 8)\n >>> x\n array([[ 0, 1, 2, 3, 4, 5, 6, 7],\n [ 8, 9, 10, 11, 12, 13, 14, 15],\n [16, 17, 18, 19, 20, 21, 22, 23]])\n\n Shuffle the rows of `x`:\n\n >>> y = rng.permuted(x, axis=1)\n >>> y\n array([[ 4, 3, 6, 7, 1, 2, 5, 0], # random\n [15, 10, 14, 9, 12, 11, 8, 13],\n [17, 16, 20, 21, 18, 22, 23, 19]])\n\n `x` has not been modified:\n\n >>> x\n array([[ 0, 1, 2, 3, 4, 5, 6, 7],\n [ 8, 9, 10, 11, 12, 13, 14, 15],\n [16, 17, 18, 19, 20, 21, 22, 23]])\n\n To shuffle the rows of `x` in-place, pass `x` as the `out`\n parameter:\n\n >>> y = rng.permuted(x, axis=1, out=x)\n >>> x\n array([[ 3, 0, 4, 7, 1, 6, 2, 5], # random\n [ 8, 14, 13, 9, 12, 11, 15, 10],\n [17, 18, 16, 22, 19, 23, 20, 21]])\n\n Note that when the ``out`` parameter is given, the return\n value is ``out``:\n\n >>> y is x\n True\n ' ... def poisson(self, lam=..., size=...) -> typing.Any: ' Draw samples from a Poisson distribution.\n\n The Poisson distribution is the limit of the binomial distribution\n for large N.\n\n Parameters\n ----------\n lam : float or array_like of floats\n Expected number of events occurring in a fixed-time interval,\n must be >= 0. A sequence must be broadcastable over the requested\n size.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``lam`` is a scalar. Otherwise,\n ``np.array(lam).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized Poisson distribution.\n\n Notes\n -----\n The Poisson distribution\n\n .. math:: f(k; \\lambda)=\\frac{\\lambda^k e^{-\\lambda}}{k!}\n\n For events with an expected separation :math:`\\lambda` the Poisson\n distribution :math:`f(k; \\lambda)` describes the probability of\n :math:`k` events occurring within the observed\n interval :math:`\\lambda`.\n\n Because the output is limited to the range of the C int64 type, a\n ValueError is raised when `lam` is within 10 sigma of the maximum\n representable value.\n\n References\n ----------\n .. [1] Weisstein, Eric W. "Poisson Distribution."\n From MathWorld--A Wolfram Web Resource.\n http://mathworld.wolfram.com/PoissonDistribution.html\n .. [2] Wikipedia, "Poisson distribution",\n https://en.wikipedia.org/wiki/Poisson_distribution\n\n Examples\n --------\n Draw samples from the distribution:\n\n >>> import numpy as np\n >>> rng = np.random.default_rng()\n >>> s = rng.poisson(5, 10000)\n\n Display histogram of the sample:\n\n >>> import matplotlib.pyplot as plt\n >>> count, bins, ignored = plt.hist(s, 14, density=True)\n >>> plt.show()\n\n Draw each 100 values for lambda 100 and 500:\n\n >>> s = rng.poisson(lam=(100., 500.), size=(100, 2))\n\n ' ... def power(self, a, size=...) -> typing.Any: ' Draws samples in [0, 1] from a power distribution with positive\n exponent a - 1.\n\n Also known as the power function distribution.\n\n Parameters\n ----------\n a : float or array_like of floats\n Parameter of the distribution. Must be non-negative.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``a`` is a scalar. Otherwise,\n ``np.array(a).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized power distribution.\n\n Raises\n ------\n ValueError\n If a <= 0.\n\n Notes\n -----\n The probability density function is\n\n .. math:: P(x; a) = ax^{a-1}, 0 \\le x \\le 1, a>0.\n\n The power function distribution is just the inverse of the Pareto\n distribution. It may also be seen as a special case of the Beta\n distribution.\n\n It is used, for example, in modeling the over-reporting of insurance\n claims.\n\n References\n ----------\n .. [1] Christian Kleiber, Samuel Kotz, "Statistical size distributions\n in economics and actuarial sciences", Wiley, 2003.\n .. [2] Heckert, N. A. and Filliben, James J. "NIST Handbook 148:\n Dataplot Reference Manual, Volume 2: Let Subcommands and Library\n Functions", National Institute of Standards and Technology\n Handbook Series, June 2003.\n https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf\n\n Examples\n --------\n Draw samples from the distribution:\n\n >>> rng = np.random.default_rng()\n >>> a = 5. # shape\n >>> samples = 1000\n >>> s = rng.power(a, samples)\n\n Display the histogram of the samples, along with\n the probability density function:\n\n >>> import matplotlib.pyplot as plt\n >>> count, bins, ignored = plt.hist(s, bins=30)\n >>> x = np.linspace(0, 1, 100)\n >>> y = a*x**(a-1.)\n >>> normed_y = samples*np.diff(bins)[0]*y\n >>> plt.plot(x, normed_y)\n >>> plt.show()\n\n Compare the power function distribution to the inverse of the Pareto.\n\n >>> from scipy import stats # doctest: +SKIP\n >>> rvs = rng.power(5, 1000000)\n >>> rvsp = rng.pareto(5, 1000000)\n >>> xx = np.linspace(0,1,100)\n >>> powpdf = stats.powerlaw.pdf(xx,5) # doctest: +SKIP\n\n >>> plt.figure()\n >>> plt.hist(rvs, bins=50, density=True)\n >>> plt.plot(xx,powpdf,\'r-\') # doctest: +SKIP\n >>> plt.title(\'power(5)\')\n\n >>> plt.figure()\n >>> plt.hist(1./(1.+rvsp), bins=50, density=True)\n >>> plt.plot(xx,powpdf,\'r-\') # doctest: +SKIP\n >>> plt.title(\'inverse of 1 + Generator.pareto(5)\')\n\n >>> plt.figure()\n >>> plt.hist(1./(1.+rvsp), bins=50, density=True)\n >>> plt.plot(xx,powpdf,\'r-\') # doctest: +SKIP\n >>> plt.title(\'inverse of stats.pareto(5)\')\n\n ' ... def random(self, size=..., dtype=..., out=...) -> typing.Any: ' Return random floats in the half-open interval [0.0, 1.0).\n\n Results are from the "continuous uniform" distribution over the\n stated interval. To sample :math:`Unif[a, b), b > a` use `uniform`\n or multiply the output of `random` by ``(b - a)`` and add ``a``::\n\n (b - a) * random() + a\n\n Parameters\n ----------\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. Default is None, in which case a\n single value is returned.\n dtype : dtype, optional\n Desired dtype of the result, only `float64` and `float32` are supported.\n Byteorder must be native. The default value is np.float64.\n out : ndarray, optional\n Alternative output array in which to place the result. If size is not None,\n it must have the same shape as the provided size and must match the type of\n the output values.\n\n Returns\n -------\n out : float or ndarray of floats\n Array of random floats of shape `size` (unless ``size=None``, in which\n case a single float is returned).\n\n See Also\n --------\n uniform : Draw samples from the parameterized uniform distribution.\n\n Examples\n --------\n >>> rng = np.random.default_rng()\n >>> rng.random()\n 0.47108547995356098 # random\n >>> type(rng.random())\n \n >>> rng.random((5,))\n array([ 0.30220482, 0.86820401, 0.1654503 , 0.11659149, 0.54323428]) # random\n\n Three-by-two array of random numbers from [-5, 0):\n\n >>> 5 * rng.random((3, 2)) - 5\n array([[-3.99149989, -0.52338984], # random\n [-2.99091858, -0.79479508],\n [-1.23204345, -1.75224494]])\n\n ' ... def rayleigh(self, scale=..., size=...) -> typing.Any: ' Draw samples from a Rayleigh distribution.\n\n The :math:`\\chi` and Weibull distributions are generalizations of the\n Rayleigh.\n\n Parameters\n ----------\n scale : float or array_like of floats, optional\n Scale, also equals the mode. Must be non-negative. Default is 1.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``scale`` is a scalar. Otherwise,\n ``np.array(scale).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized Rayleigh distribution.\n\n Notes\n -----\n The probability density function for the Rayleigh distribution is\n\n .. math:: P(x;scale) = \\frac{x}{scale^2}e^{\\frac{-x^2}{2 \\cdotp scale^2}}\n\n The Rayleigh distribution would arise, for example, if the East\n and North components of the wind velocity had identical zero-mean\n Gaussian distributions. Then the wind speed would have a Rayleigh\n distribution.\n\n References\n ----------\n .. [1] Brighton Webs Ltd., "Rayleigh Distribution,"\n https://web.archive.org/web/20090514091424/http://brighton-webs.co.uk:80/distributions/rayleigh.asp\n .. [2] Wikipedia, "Rayleigh distribution"\n https://en.wikipedia.org/wiki/Rayleigh_distribution\n\n Examples\n --------\n Draw values from the distribution and plot the histogram\n\n >>> from matplotlib.pyplot import hist\n >>> rng = np.random.default_rng()\n >>> values = hist(rng.rayleigh(3, 100000), bins=200, density=True)\n\n Wave heights tend to follow a Rayleigh distribution. If the mean wave\n height is 1 meter, what fraction of waves are likely to be larger than 3\n meters?\n\n >>> meanvalue = 1\n >>> modevalue = np.sqrt(2 / np.pi) * meanvalue\n >>> s = rng.rayleigh(modevalue, 1000000)\n\n The percentage of waves larger than 3 meters is:\n\n >>> 100.*sum(s>3)/1000000.\n 0.087300000000000003 # random\n\n ' ... def shuffle(self, x, axis=...) -> typing.Any: ' Modify an array or sequence in-place by shuffling its contents.\n\n The order of sub-arrays is changed but their contents remains the same.\n\n Parameters\n ----------\n x : ndarray or MutableSequence\n The array, list or mutable sequence to be shuffled.\n axis : int, optional\n The axis which `x` is shuffled along. Default is 0.\n It is only supported on `ndarray` objects.\n\n Returns\n -------\n None\n\n See Also\n --------\n permuted\n permutation\n\n Notes\n -----\n An important distinction between methods ``shuffle`` and ``permuted`` is \n how they both treat the ``axis`` parameter which can be found at \n :ref:`generator-handling-axis-parameter`.\n\n Examples\n --------\n >>> rng = np.random.default_rng()\n >>> arr = np.arange(10)\n >>> arr\n array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])\n >>> rng.shuffle(arr)\n >>> arr\n array([2, 0, 7, 5, 1, 4, 8, 9, 3, 6]) # random\n\n >>> arr = np.arange(9).reshape((3, 3))\n >>> arr\n array([[0, 1, 2],\n [3, 4, 5],\n [6, 7, 8]])\n >>> rng.shuffle(arr)\n >>> arr\n array([[3, 4, 5], # random\n [6, 7, 8],\n [0, 1, 2]])\n\n >>> arr = np.arange(9).reshape((3, 3))\n >>> arr\n array([[0, 1, 2],\n [3, 4, 5],\n [6, 7, 8]])\n >>> rng.shuffle(arr, axis=1)\n >>> arr\n array([[2, 0, 1], # random\n [5, 3, 4],\n [8, 6, 7]])\n ' ... def spawn(self, n_children) -> typing.Any: ' Create new independent child generators.\n\n See :ref:`seedsequence-spawn` for additional notes on spawning\n children.\n\n .. versionadded:: 1.25.0\n\n Parameters\n ----------\n n_children : int\n\n Returns\n -------\n child_generators : list of Generators\n\n Raises\n ------\n TypeError\n When the underlying SeedSequence does not implement spawning.\n\n See Also\n --------\n random.BitGenerator.spawn, random.SeedSequence.spawn :\n Equivalent method on the bit generator and seed sequence.\n bit_generator :\n The bit generator instance used by the generator.\n\n Examples\n --------\n Starting from a seeded default generator:\n\n >>> # High quality entropy created with: f"0x{secrets.randbits(128):x}"\n >>> entropy = 0x3034c61a9ae04ff8cb62ab8ec2c4b501\n >>> rng = np.random.default_rng(entropy)\n\n Create two new generators for example for parallel execution:\n\n >>> child_rng1, child_rng2 = rng.spawn(2)\n\n Drawn numbers from each are independent but derived from the initial\n seeding entropy:\n\n >>> rng.uniform(), child_rng1.uniform(), child_rng2.uniform()\n (0.19029263503854454, 0.9475673279178444, 0.4702687338396767)\n\n It is safe to spawn additional children from the original ``rng`` or\n the children:\n\n >>> more_child_rngs = rng.spawn(20)\n >>> nested_spawn = child_rng1.spawn(20)\n\n ' ... def standard_cauchy(self, size=...) -> typing.Any: ' Draw samples from a standard Cauchy distribution with mode = 0.\n\n Also known as the Lorentz distribution.\n\n Parameters\n ----------\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. Default is None, in which case a\n single value is returned.\n\n Returns\n -------\n samples : ndarray or scalar\n The drawn samples.\n\n Notes\n -----\n The probability density function for the full Cauchy distribution is\n\n .. math:: P(x; x_0, \\gamma) = \\frac{1}{\\pi \\gamma \\bigl[ 1+\n (\\frac{x-x_0}{\\gamma})^2 \\bigr] }\n\n and the Standard Cauchy distribution just sets :math:`x_0=0` and\n :math:`\\gamma=1`\n\n The Cauchy distribution arises in the solution to the driven harmonic\n oscillator problem, and also describes spectral line broadening. It\n also describes the distribution of values at which a line tilted at\n a random angle will cut the x axis.\n\n When studying hypothesis tests that assume normality, seeing how the\n tests perform on data from a Cauchy distribution is a good indicator of\n their sensitivity to a heavy-tailed distribution, since the Cauchy looks\n very much like a Gaussian distribution, but with heavier tails.\n\n References\n ----------\n .. [1] NIST/SEMATECH e-Handbook of Statistical Methods, "Cauchy\n Distribution",\n https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm\n .. [2] Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A\n Wolfram Web Resource.\n http://mathworld.wolfram.com/CauchyDistribution.html\n .. [3] Wikipedia, "Cauchy distribution"\n https://en.wikipedia.org/wiki/Cauchy_distribution\n\n Examples\n --------\n Draw samples and plot the distribution:\n\n >>> import matplotlib.pyplot as plt\n >>> s = np.random.default_rng().standard_cauchy(1000000)\n >>> s = s[(s>-25) & (s<25)] # truncate distribution so it plots well\n >>> plt.hist(s, bins=100)\n >>> plt.show()\n\n ' ... def standard_exponential(self, size=..., dtype=..., method=..., out=...) -> typing.Any: " Draw samples from the standard exponential distribution.\n\n `standard_exponential` is identical to the exponential distribution\n with a scale parameter of 1.\n\n Parameters\n ----------\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. Default is None, in which case a\n single value is returned.\n dtype : dtype, optional\n Desired dtype of the result, only `float64` and `float32` are supported.\n Byteorder must be native. The default value is np.float64.\n method : str, optional\n Either 'inv' or 'zig'. 'inv' uses the default inverse CDF method.\n 'zig' uses the much faster Ziggurat method of Marsaglia and Tsang.\n out : ndarray, optional\n Alternative output array in which to place the result. If size is not None,\n it must have the same shape as the provided size and must match the type of\n the output values.\n\n Returns\n -------\n out : float or ndarray\n Drawn samples.\n\n Examples\n --------\n Output a 3x8000 array:\n\n >>> n = np.random.default_rng().standard_exponential((3, 8000))\n\n " ... def standard_gamma(self, shape, size=..., dtype=..., out=...) -> typing.Any: ' Draw samples from a standard Gamma distribution.\n\n Samples are drawn from a Gamma distribution with specified parameters,\n shape (sometimes designated "k") and scale=1.\n\n Parameters\n ----------\n shape : float or array_like of floats\n Parameter, must be non-negative.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``shape`` is a scalar. Otherwise,\n ``np.array(shape).size`` samples are drawn.\n dtype : dtype, optional\n Desired dtype of the result, only `float64` and `float32` are supported.\n Byteorder must be native. The default value is np.float64.\n out : ndarray, optional\n Alternative output array in which to place the result. If size is\n not None, it must have the same shape as the provided size and\n must match the type of the output values.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized standard gamma distribution.\n\n See Also\n --------\n scipy.stats.gamma : probability density function, distribution or\n cumulative density function, etc.\n\n Notes\n -----\n The probability density for the Gamma distribution is\n\n .. math:: p(x) = x^{k-1}\\frac{e^{-x/\\theta}}{\\theta^k\\Gamma(k)},\n\n where :math:`k` is the shape and :math:`\\theta` the scale,\n and :math:`\\Gamma` is the Gamma function.\n\n The Gamma distribution is often used to model the times to failure of\n electronic components, and arises naturally in processes for which the\n waiting times between Poisson distributed events are relevant.\n\n References\n ----------\n .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A\n Wolfram Web Resource.\n http://mathworld.wolfram.com/GammaDistribution.html\n .. [2] Wikipedia, "Gamma distribution",\n https://en.wikipedia.org/wiki/Gamma_distribution\n\n Examples\n --------\n Draw samples from the distribution:\n\n >>> shape, scale = 2., 1. # mean and width\n >>> s = np.random.default_rng().standard_gamma(shape, 1000000)\n\n Display the histogram of the samples, along with\n the probability density function:\n\n >>> import matplotlib.pyplot as plt\n >>> import scipy.special as sps # doctest: +SKIP\n >>> count, bins, ignored = plt.hist(s, 50, density=True)\n >>> y = bins**(shape-1) * ((np.exp(-bins/scale))/ # doctest: +SKIP\n ... (sps.gamma(shape) * scale**shape))\n >>> plt.plot(bins, y, linewidth=2, color=\'r\') # doctest: +SKIP\n >>> plt.show()\n\n ' ... def standard_normal(self, size=..., dtype=..., out=...) -> typing.Any: ' Draw samples from a standard Normal distribution (mean=0, stdev=1).\n\n Parameters\n ----------\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. Default is None, in which case a\n single value is returned.\n dtype : dtype, optional\n Desired dtype of the result, only `float64` and `float32` are supported.\n Byteorder must be native. The default value is np.float64.\n out : ndarray, optional\n Alternative output array in which to place the result. If size is not None,\n it must have the same shape as the provided size and must match the type of\n the output values.\n\n Returns\n -------\n out : float or ndarray\n A floating-point array of shape ``size`` of drawn samples, or a\n single sample if ``size`` was not specified.\n\n See Also\n --------\n normal :\n Equivalent function with additional ``loc`` and ``scale`` arguments\n for setting the mean and standard deviation.\n\n Notes\n -----\n For random samples from the normal distribution with mean ``mu`` and\n standard deviation ``sigma``, use one of::\n\n mu + sigma * rng.standard_normal(size=...)\n rng.normal(mu, sigma, size=...)\n\n Examples\n --------\n >>> rng = np.random.default_rng()\n >>> rng.standard_normal()\n 2.1923875335537315 # random\n\n >>> s = rng.standard_normal(8000)\n >>> s\n array([ 0.6888893 , 0.78096262, -0.89086505, ..., 0.49876311, # random\n -0.38672696, -0.4685006 ]) # random\n >>> s.shape\n (8000,)\n >>> s = rng.standard_normal(size=(3, 4, 2))\n >>> s.shape\n (3, 4, 2)\n\n Two-by-four array of samples from the normal distribution with\n mean 3 and standard deviation 2.5:\n\n >>> 3 + 2.5 * rng.standard_normal(size=(2, 4))\n array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random\n [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random\n\n ' ... def standard_t(self, df, size=...) -> typing.Any: ' Draw samples from a standard Student\'s t distribution with `df` degrees\n of freedom.\n\n A special case of the hyperbolic distribution. As `df` gets\n large, the result resembles that of the standard normal\n distribution (`standard_normal`).\n\n Parameters\n ----------\n df : float or array_like of floats\n Degrees of freedom, must be > 0.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``df`` is a scalar. Otherwise,\n ``np.array(df).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized standard Student\'s t distribution.\n\n Notes\n -----\n The probability density function for the t distribution is\n\n .. math:: P(x, df) = \\frac{\\Gamma(\\frac{df+1}{2})}{\\sqrt{\\pi df}\n \\Gamma(\\frac{df}{2})}\\Bigl( 1+\\frac{x^2}{df} \\Bigr)^{-(df+1)/2}\n\n The t test is based on an assumption that the data come from a\n Normal distribution. The t test provides a way to test whether\n the sample mean (that is the mean calculated from the data) is\n a good estimate of the true mean.\n\n The derivation of the t-distribution was first published in\n 1908 by William Gosset while working for the Guinness Brewery\n in Dublin. Due to proprietary issues, he had to publish under\n a pseudonym, and so he used the name Student.\n\n References\n ----------\n .. [1] Dalgaard, Peter, "Introductory Statistics With R",\n Springer, 2002.\n .. [2] Wikipedia, "Student\'s t-distribution"\n https://en.wikipedia.org/wiki/Student\'s_t-distribution\n\n Examples\n --------\n From Dalgaard page 83 [1]_, suppose the daily energy intake for 11\n women in kilojoules (kJ) is:\n\n >>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \\\n ... 7515, 8230, 8770])\n\n Does their energy intake deviate systematically from the recommended\n value of 7725 kJ? Our null hypothesis will be the absence of deviation,\n and the alternate hypothesis will be the presence of an effect that could be\n either positive or negative, hence making our test 2-tailed. \n\n Because we are estimating the mean and we have N=11 values in our sample,\n we have N-1=10 degrees of freedom. We set our significance level to 95% and \n compute the t statistic using the empirical mean and empirical standard \n deviation of our intake. We use a ddof of 1 to base the computation of our \n empirical standard deviation on an unbiased estimate of the variance (note:\n the final estimate is not unbiased due to the concave nature of the square \n root).\n\n >>> np.mean(intake)\n 6753.636363636364\n >>> intake.std(ddof=1)\n 1142.1232221373727\n >>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake)))\n >>> t\n -2.8207540608310198\n\n We draw 1000000 samples from Student\'s t distribution with the adequate\n degrees of freedom.\n\n >>> import matplotlib.pyplot as plt\n >>> s = np.random.default_rng().standard_t(10, size=1000000)\n >>> h = plt.hist(s, bins=100, density=True)\n\n Does our t statistic land in one of the two critical regions found at \n both tails of the distribution?\n\n >>> np.sum(np.abs(t) < np.abs(s)) / float(len(s))\n 0.018318 #random < 0.05, statistic is in critical region\n\n The probability value for this 2-tailed test is about 1.83%, which is \n lower than the 5% pre-determined significance threshold. \n\n Therefore, the probability of observing values as extreme as our intake\n conditionally on the null hypothesis being true is too low, and we reject \n the null hypothesis of no deviation. \n\n ' ... def triangular(self, left, mode, right, size=...) -> typing.Any: ' Draw samples from the triangular distribution over the\n interval ``[left, right]``.\n\n The triangular distribution is a continuous probability\n distribution with lower limit left, peak at mode, and upper\n limit right. Unlike the other distributions, these parameters\n directly define the shape of the pdf.\n\n Parameters\n ----------\n left : float or array_like of floats\n Lower limit.\n mode : float or array_like of floats\n The value where the peak of the distribution occurs.\n The value must fulfill the condition ``left <= mode <= right``.\n right : float or array_like of floats\n Upper limit, must be larger than `left`.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``left``, ``mode``, and ``right``\n are all scalars. Otherwise, ``np.broadcast(left, mode, right).size``\n samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized triangular distribution.\n\n Notes\n -----\n The probability density function for the triangular distribution is\n\n .. math:: P(x;l, m, r) = \\begin{cases}\n \\frac{2(x-l)}{(r-l)(m-l)}& \\text{for $l \\leq x \\leq m$},\\\\\n \\frac{2(r-x)}{(r-l)(r-m)}& \\text{for $m \\leq x \\leq r$},\\\\\n 0& \\text{otherwise}.\n \\end{cases}\n\n The triangular distribution is often used in ill-defined\n problems where the underlying distribution is not known, but\n some knowledge of the limits and mode exists. Often it is used\n in simulations.\n\n References\n ----------\n .. [1] Wikipedia, "Triangular distribution"\n https://en.wikipedia.org/wiki/Triangular_distribution\n\n Examples\n --------\n Draw values from the distribution and plot the histogram:\n\n >>> import matplotlib.pyplot as plt\n >>> h = plt.hist(np.random.default_rng().triangular(-3, 0, 8, 100000), bins=200,\n ... density=True)\n >>> plt.show()\n\n ' ... def uniform(self, low=..., high=..., size=...) -> typing.Any: " Draw samples from a uniform distribution.\n\n Samples are uniformly distributed over the half-open interval\n ``[low, high)`` (includes low, but excludes high). In other words,\n any value within the given interval is equally likely to be drawn\n by `uniform`.\n\n Parameters\n ----------\n low : float or array_like of floats, optional\n Lower boundary of the output interval. All values generated will be\n greater than or equal to low. The default value is 0.\n high : float or array_like of floats\n Upper boundary of the output interval. All values generated will be\n less than high. The high limit may be included in the returned array of \n floats due to floating-point rounding in the equation \n ``low + (high-low) * random_sample()``. high - low must be \n non-negative. The default value is 1.0.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``low`` and ``high`` are both scalars.\n Otherwise, ``np.broadcast(low, high).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized uniform distribution.\n\n See Also\n --------\n integers : Discrete uniform distribution, yielding integers.\n random : Floats uniformly distributed over ``[0, 1)``.\n\n Notes\n -----\n The probability density function of the uniform distribution is\n\n .. math:: p(x) = \\frac{1}{b - a}\n\n anywhere within the interval ``[a, b)``, and zero elsewhere.\n\n When ``high`` == ``low``, values of ``low`` will be returned.\n\n Examples\n --------\n Draw samples from the distribution:\n\n >>> s = np.random.default_rng().uniform(-1,0,1000)\n\n All values are within the given interval:\n\n >>> np.all(s >= -1)\n True\n >>> np.all(s < 0)\n True\n\n Display the histogram of the samples, along with the\n probability density function:\n\n >>> import matplotlib.pyplot as plt\n >>> count, bins, ignored = plt.hist(s, 15, density=True)\n >>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r')\n >>> plt.show()\n\n " ... def vonmises(self, mu, kappa, size=...) -> typing.Any: ' Draw samples from a von Mises distribution.\n\n Samples are drawn from a von Mises distribution with specified mode\n (mu) and dispersion (kappa), on the interval [-pi, pi].\n\n The von Mises distribution (also known as the circular normal\n distribution) is a continuous probability distribution on the unit\n circle. It may be thought of as the circular analogue of the normal\n distribution.\n\n Parameters\n ----------\n mu : float or array_like of floats\n Mode ("center") of the distribution.\n kappa : float or array_like of floats\n Dispersion of the distribution, has to be >=0.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``mu`` and ``kappa`` are both scalars.\n Otherwise, ``np.broadcast(mu, kappa).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized von Mises distribution.\n\n See Also\n --------\n scipy.stats.vonmises : probability density function, distribution, or\n cumulative density function, etc.\n\n Notes\n -----\n The probability density for the von Mises distribution is\n\n .. math:: p(x) = \\frac{e^{\\kappa cos(x-\\mu)}}{2\\pi I_0(\\kappa)},\n\n where :math:`\\mu` is the mode and :math:`\\kappa` the dispersion,\n and :math:`I_0(\\kappa)` is the modified Bessel function of order 0.\n\n The von Mises is named for Richard Edler von Mises, who was born in\n Austria-Hungary, in what is now the Ukraine. He fled to the United\n States in 1939 and became a professor at Harvard. He worked in\n probability theory, aerodynamics, fluid mechanics, and philosophy of\n science.\n\n References\n ----------\n .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of\n Mathematical Functions with Formulas, Graphs, and Mathematical\n Tables, 9th printing," New York: Dover, 1972.\n .. [2] von Mises, R., "Mathematical Theory of Probability\n and Statistics", New York: Academic Press, 1964.\n\n Examples\n --------\n Draw samples from the distribution:\n\n >>> mu, kappa = 0.0, 4.0 # mean and dispersion\n >>> s = np.random.default_rng().vonmises(mu, kappa, 1000)\n\n Display the histogram of the samples, along with\n the probability density function:\n\n >>> import matplotlib.pyplot as plt\n >>> from scipy.special import i0 # doctest: +SKIP\n >>> plt.hist(s, 50, density=True)\n >>> x = np.linspace(-np.pi, np.pi, num=51)\n >>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa)) # doctest: +SKIP\n >>> plt.plot(x, y, linewidth=2, color=\'r\') # doctest: +SKIP\n >>> plt.show()\n\n ' ... def wald(self, mean, scale, size=...) -> typing.Any: ' Draw samples from a Wald, or inverse Gaussian, distribution.\n\n As the scale approaches infinity, the distribution becomes more like a\n Gaussian. Some references claim that the Wald is an inverse Gaussian\n with mean equal to 1, but this is by no means universal.\n\n The inverse Gaussian distribution was first studied in relationship to\n Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian\n because there is an inverse relationship between the time to cover a\n unit distance and distance covered in unit time.\n\n Parameters\n ----------\n mean : float or array_like of floats\n Distribution mean, must be > 0.\n scale : float or array_like of floats\n Scale parameter, must be > 0.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``mean`` and ``scale`` are both scalars.\n Otherwise, ``np.broadcast(mean, scale).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized Wald distribution.\n\n Notes\n -----\n The probability density function for the Wald distribution is\n\n .. math:: P(x;mean,scale) = \\sqrt{\\frac{scale}{2\\pi x^3}}e^\n \\frac{-scale(x-mean)^2}{2\\cdotp mean^2x}\n\n As noted above the inverse Gaussian distribution first arise\n from attempts to model Brownian motion. It is also a\n competitor to the Weibull for use in reliability modeling and\n modeling stock returns and interest rate processes.\n\n References\n ----------\n .. [1] Brighton Webs Ltd., Wald Distribution,\n https://web.archive.org/web/20090423014010/http://www.brighton-webs.co.uk:80/distributions/wald.asp\n .. [2] Chhikara, Raj S., and Folks, J. Leroy, "The Inverse Gaussian\n Distribution: Theory : Methodology, and Applications", CRC Press,\n 1988.\n .. [3] Wikipedia, "Inverse Gaussian distribution"\n https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution\n\n Examples\n --------\n Draw values from the distribution and plot the histogram:\n\n >>> import matplotlib.pyplot as plt\n >>> h = plt.hist(np.random.default_rng().wald(3, 2, 100000), bins=200, density=True)\n >>> plt.show()\n\n ' ... def weibull(self, a, size=...) -> typing.Any: ' Draw samples from a Weibull distribution.\n\n Draw samples from a 1-parameter Weibull distribution with the given\n shape parameter `a`.\n\n .. math:: X = (-ln(U))^{1/a}\n\n Here, U is drawn from the uniform distribution over (0,1].\n\n The more common 2-parameter Weibull, including a scale parameter\n :math:`\\lambda` is just :math:`X = \\lambda(-ln(U))^{1/a}`.\n\n Parameters\n ----------\n a : float or array_like of floats\n Shape parameter of the distribution. Must be nonnegative.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``a`` is a scalar. Otherwise,\n ``np.array(a).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized Weibull distribution.\n\n See Also\n --------\n scipy.stats.weibull_max\n scipy.stats.weibull_min\n scipy.stats.genextreme\n gumbel\n\n Notes\n -----\n The Weibull (or Type III asymptotic extreme value distribution\n for smallest values, SEV Type III, or Rosin-Rammler\n distribution) is one of a class of Generalized Extreme Value\n (GEV) distributions used in modeling extreme value problems.\n This class includes the Gumbel and Frechet distributions.\n\n The probability density for the Weibull distribution is\n\n .. math:: p(x) = \\frac{a}\n {\\lambda}(\\frac{x}{\\lambda})^{a-1}e^{-(x/\\lambda)^a},\n\n where :math:`a` is the shape and :math:`\\lambda` the scale.\n\n The function has its peak (the mode) at\n :math:`\\lambda(\\frac{a-1}{a})^{1/a}`.\n\n When ``a = 1``, the Weibull distribution reduces to the exponential\n distribution.\n\n References\n ----------\n .. [1] Waloddi Weibull, Royal Technical University, Stockholm,\n 1939 "A Statistical Theory Of The Strength Of Materials",\n Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939,\n Generalstabens Litografiska Anstalts Forlag, Stockholm.\n .. [2] Waloddi Weibull, "A Statistical Distribution Function of\n Wide Applicability", Journal Of Applied Mechanics ASME Paper\n 1951.\n .. [3] Wikipedia, "Weibull distribution",\n https://en.wikipedia.org/wiki/Weibull_distribution\n\n Examples\n --------\n Draw samples from the distribution:\n\n >>> rng = np.random.default_rng()\n >>> a = 5. # shape\n >>> s = rng.weibull(a, 1000)\n\n Display the histogram of the samples, along with\n the probability density function:\n\n >>> import matplotlib.pyplot as plt\n >>> x = np.arange(1,100.)/50.\n >>> def weib(x,n,a):\n ... return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)\n\n >>> count, bins, ignored = plt.hist(rng.weibull(5.,1000))\n >>> x = np.arange(1,100.)/50.\n >>> scale = count.max()/weib(x, 1., 5.).max()\n >>> plt.plot(x, weib(x, 1., 5.)*scale)\n >>> plt.show()\n\n ' ... def zipf(self, a, size=...) -> typing.Any: ' Draw samples from a Zipf distribution.\n\n Samples are drawn from a Zipf distribution with specified parameter\n `a` > 1.\n\n The Zipf distribution (also known as the zeta distribution) is a\n discrete probability distribution that satisfies Zipf\'s law: the\n frequency of an item is inversely proportional to its rank in a\n frequency table.\n\n Parameters\n ----------\n a : float or array_like of floats\n Distribution parameter. Must be greater than 1.\n size : int or tuple of ints, optional\n Output shape. If the given shape is, e.g., ``(m, n, k)``, then\n ``m * n * k`` samples are drawn. If size is ``None`` (default),\n a single value is returned if ``a`` is a scalar. Otherwise,\n ``np.array(a).size`` samples are drawn.\n\n Returns\n -------\n out : ndarray or scalar\n Drawn samples from the parameterized Zipf distribution.\n\n See Also\n --------\n scipy.stats.zipf : probability density function, distribution, or\n cumulative density function, etc.\n\n Notes\n -----\n The probability density for the Zipf distribution is\n\n .. math:: p(k) = \\frac{k^{-a}}{\\zeta(a)},\n\n for integers :math:`k \\geq 1`, where :math:`\\zeta` is the Riemann Zeta\n function.\n\n It is named for the American linguist George Kingsley Zipf, who noted\n that the frequency of any word in a sample of a language is inversely\n proportional to its rank in the frequency table.\n\n References\n ----------\n .. [1] Zipf, G. K., "Selected Studies of the Principle of Relative\n Frequency in Language," Cambridge, MA: Harvard Univ. Press,\n 1932.\n\n Examples\n --------\n Draw samples from the distribution:\n\n >>> a = 4.0\n >>> n = 20000\n >>> s = np.random.default_rng().zipf(a, size=n)\n\n Display the histogram of the samples, along with\n the expected histogram based on the probability\n density function:\n\n >>> import matplotlib.pyplot as plt\n >>> from scipy.special import zeta # doctest: +SKIP\n\n `bincount` provides a fast histogram for small integers.\n\n >>> count = np.bincount(s)\n >>> k = np.arange(1, s.max() + 1)\n\n >>> plt.bar(k, count[1:], alpha=0.5, label=\'sample count\')\n >>> plt.plot(k, n*(k**-a)/zeta(a), \'k.-\', alpha=0.5,\n ... label=\'expected count\') # doctest: +SKIP\n >>> plt.semilogy()\n >>> plt.grid(alpha=0.4)\n >>> plt.legend()\n >>> plt.title(f\'Zipf sample, a={a}, size={n}\')\n >>> plt.show()\n\n ' ... def __getattr__(self, name) -> typing.Any: ... PCG64 = _mod_numpy_random__pcg64.PCG64 Sequence = _mod_collections_abc.Sequence __doc__: typing.Any __file__: str __name__: str __package__: str def __pyx_unpickle_Enum() -> typing.Any: ... __test__: dict def default_rng(seed=...) -> typing.Any: "Construct a new Generator with the default BitGenerator (PCG64).\n\n Parameters\n ----------\n seed : {None, int, array_like[ints], SeedSequence, BitGenerator, Generator}, optional\n A seed to initialize the `BitGenerator`. If None, then fresh,\n unpredictable entropy will be pulled from the OS. If an ``int`` or\n ``array_like[ints]`` is passed, then it will be passed to\n `SeedSequence` to derive the initial `BitGenerator` state. One may also\n pass in a `SeedSequence` instance.\n Additionally, when passed a `BitGenerator`, it will be wrapped by\n `Generator`. If passed a `Generator`, it will be returned unaltered.\n\n Returns\n -------\n Generator\n The initialized generator object.\n\n Notes\n -----\n If ``seed`` is not a `BitGenerator` or a `Generator`, a new `BitGenerator`\n is instantiated. This function does not manage a default global instance.\n\n See :ref:`seeding_and_entropy` for more information about seeding.\n \n Examples\n --------\n ``default_rng`` is the recommended constructor for the random number class\n ``Generator``. Here are several ways we can construct a random \n number generator using ``default_rng`` and the ``Generator`` class. \n \n Here we use ``default_rng`` to generate a random float:\n \n >>> import numpy as np\n >>> rng = np.random.default_rng(12345)\n >>> print(rng)\n Generator(PCG64)\n >>> rfloat = rng.random()\n >>> rfloat\n 0.22733602246716966\n >>> type(rfloat)\n \n \n Here we use ``default_rng`` to generate 3 random integers between 0 \n (inclusive) and 10 (exclusive):\n \n >>> import numpy as np\n >>> rng = np.random.default_rng(12345)\n >>> rints = rng.integers(low=0, high=10, size=3)\n >>> rints\n array([6, 2, 7])\n >>> type(rints[0])\n \n \n Here we specify a seed so that we have reproducible results:\n \n >>> import numpy as np\n >>> rng = np.random.default_rng(seed=42)\n >>> print(rng)\n Generator(PCG64)\n >>> arr1 = rng.random((3, 3))\n >>> arr1\n array([[0.77395605, 0.43887844, 0.85859792],\n [0.69736803, 0.09417735, 0.97562235],\n [0.7611397 , 0.78606431, 0.12811363]])\n\n If we exit and restart our Python interpreter, we'll see that we\n generate the same random numbers again:\n\n >>> import numpy as np\n >>> rng = np.random.default_rng(seed=42)\n >>> arr2 = rng.random((3, 3))\n >>> arr2\n array([[0.77395605, 0.43887844, 0.85859792],\n [0.69736803, 0.09417735, 0.97562235],\n [0.7611397 , 0.78606431, 0.12811363]])\n\n " ... def normalize_axis_index(axis, ndim, msg_prefix=...) -> typing.Any: " Normalizes an axis index, `axis`, such that is a valid positive index into\n the shape of array with `ndim` dimensions. Raises an AxisError with an\n appropriate message if this is not possible.\n\n Used internally by all axis-checking logic.\n\n .. versionadded:: 1.13.0\n\n Parameters\n ----------\n axis : int\n The un-normalized index of the axis. Can be negative\n ndim : int\n The number of dimensions of the array that `axis` should be normalized\n against\n msg_prefix : str\n A prefix to put before the message, typically the name of the argument\n\n Returns\n -------\n normalized_axis : int\n The normalized axis index, such that `0 <= normalized_axis < ndim`\n\n Raises\n ------\n AxisError\n If the axis index is invalid, when `-ndim <= axis < ndim` is false.\n\n Examples\n --------\n >>> normalize_axis_index(0, ndim=3)\n 0\n >>> normalize_axis_index(1, ndim=3)\n 1\n >>> normalize_axis_index(-1, ndim=3)\n 2\n\n >>> normalize_axis_index(3, ndim=3)\n Traceback (most recent call last):\n ...\n AxisError: axis 3 is out of bounds for array of dimension 3\n >>> normalize_axis_index(-4, ndim=3, msg_prefix='axes_arg')\n Traceback (most recent call last):\n ...\n AxisError: axes_arg: axis -4 is out of bounds for array of dimension 3" ... def __getattr__(name) -> typing.Any: ...