scipy.stats.yulesimon = <scipy.stats._discrete_distns.yulesimon_gen object>[source]#

A Yule-Simon discrete random variable.

As an instance of the rv_discrete class, yulesimon object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.


The probability mass function for the yulesimon is:

\[f(k) = \alpha B(k, \alpha+1)\]

for \(k=1,2,3,...\), where \(\alpha>0\). Here \(B\) refers to the scipy.special.beta function.

The sampling of random variates is based on pg 553, Section 6.3 of [1]. Our notation maps to the referenced logic via \(\alpha=a-1\).

For details see the wikipedia entry [2].



Devroye, Luc. “Non-uniform Random Variate Generation”, (1986) Springer, New York.

The probability mass function above is defined in the “standardized” form. To shift distribution use the loc parameter. Specifically, yulesimon.pmf(k, alpha, loc) is identically equivalent to yulesimon.pmf(k - loc, alpha).


>>> import numpy as np
>>> from scipy.stats import yulesimon
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

>>> alpha = 11
>>> mean, var, skew, kurt = yulesimon.stats(alpha, moments='mvsk')

Display the probability mass function (pmf):

>>> x = np.arange(yulesimon.ppf(0.01, alpha),
...               yulesimon.ppf(0.99, alpha))
>>> ax.plot(x, yulesimon.pmf(x, alpha), 'bo', ms=8, label='yulesimon pmf')
>>> ax.vlines(x, 0, yulesimon.pmf(x, alpha), colors='b', lw=5, alpha=0.5)

Alternatively, the distribution object can be called (as a function) to fix the shape and location. This returns a “frozen” RV object holding the given parameters fixed.

Freeze the distribution and display the frozen pmf:

>>> rv = yulesimon(alpha)
>>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1,
...         label='frozen pmf')
>>> ax.legend(loc='best', frameon=False)

Check accuracy of cdf and ppf:

>>> prob = yulesimon.cdf(x, alpha)
>>> np.allclose(x, yulesimon.ppf(prob, alpha))

Generate random numbers:

>>> r = yulesimon.rvs(alpha, size=1000)


rvs(alpha, loc=0, size=1, random_state=None)

Random variates.

pmf(k, alpha, loc=0)

Probability mass function.

logpmf(k, alpha, loc=0)

Log of the probability mass function.

cdf(k, alpha, loc=0)

Cumulative distribution function.

logcdf(k, alpha, loc=0)

Log of the cumulative distribution function.

sf(k, alpha, loc=0)

Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).

logsf(k, alpha, loc=0)

Log of the survival function.

ppf(q, alpha, loc=0)

Percent point function (inverse of cdf — percentiles).

isf(q, alpha, loc=0)

Inverse survival function (inverse of sf).

stats(alpha, loc=0, moments=’mv’)

Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).

entropy(alpha, loc=0)

(Differential) entropy of the RV.

expect(func, args=(alpha,), loc=0, lb=None, ub=None, conditional=False)

Expected value of a function (of one argument) with respect to the distribution.

median(alpha, loc=0)

Median of the distribution.

mean(alpha, loc=0)

Mean of the distribution.

var(alpha, loc=0)

Variance of the distribution.

std(alpha, loc=0)

Standard deviation of the distribution.

interval(confidence, alpha, loc=0)

Confidence interval with equal areas around the median.