# scipy.stats.yulesimon

scipy.stats.yulesimon = <scipy.stats._discrete_distns.yulesimon_gen object>

  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.

  Notes

  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].

  References

  [1]

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

  [2]

  https://en.wikipedia.org/wiki/Yule-Simon_distribution

  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)`.

  Examples

      >>> 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)
      >>> plt.show()

  Check accuracy of `cdf` and `ppf`:

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

  Generate random numbers:

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

  Methods

  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.