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# scipy.stats.betanbinom

scipy.stats.betanbinom = <scipy.stats._discrete_distns.betanbinom_gen object>

  A beta-negative-binomial discrete random variable.

  As an instance of the `rv_discrete` class, `betanbinom` 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.

  See also

  `betabinom`

    Beta binomial distribution

  Notes

  The beta-negative-binomial distribution is a negative binomial distribution with a probability of success p that follows a beta distribution.

  The probability mass function for `betanbinom` is:

  \[f(k) = \binom{n + k - 1}{k} \frac{B(a + n, b + k)}{B(a, b)}\]

  for \(k \ge 0\), \(n \geq 0\), \(a > 0\), \(b > 0\), where \(B(a, b)\) is the beta function.

  `betanbinom` takes \(n\), \(a\), and \(b\) as shape parameters.

  References

  [1]

  https://en.wikipedia.org/wiki/Beta_negative_binomial_distribution

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

  New in version 1.12.0.

  Examples

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

  Calculate the first four moments:

      >>> n, a, b = 5, 9.3, 1
      >>> mean, var, skew, kurt = betanbinom.stats(n, a, b, moments='mvsk')

  Display the probability mass function (`pmf`):

      >>> x = np.arange(betanbinom.ppf(0.01, n, a, b),
      ...               betanbinom.ppf(0.99, n, a, b))
      >>> ax.plot(x, betanbinom.pmf(x, n, a, b), 'bo', ms=8, label='betanbinom pmf')
      >>> ax.vlines(x, 0, betanbinom.pmf(x, n, a, b), 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 = betanbinom(n, a, b)
      >>> 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 = betanbinom.cdf(x, n, a, b)
      >>> np.allclose(x, betanbinom.ppf(prob, n, a, b))
      True

  Generate random numbers:

      >>> r = betanbinom.rvs(n, a, b, size=1000)

  Methods

  rvs(n, a, b, loc=0, size=1, random_state=None) | Random variates.
  pmf(k, n, a, b, loc=0) | Probability mass function.
  logpmf(k, n, a, b, loc=0) | Log of the probability mass function.
  cdf(k, n, a, b, loc=0) | Cumulative distribution function.
  logcdf(k, n, a, b, loc=0) | Log of the cumulative distribution function.
  sf(k, n, a, b, loc=0) | Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
  logsf(k, n, a, b, loc=0) | Log of the survival function.
  ppf(q, n, a, b, loc=0) | Percent point function (inverse of `cdf` — percentiles).
  isf(q, n, a, b, loc=0) | Inverse survival function (inverse of `sf`).
  stats(n, a, b, loc=0, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
  entropy(n, a, b, loc=0) | (Differential) entropy of the RV.
  expect(func, args=(n, a, b), loc=0, lb=None, ub=None, conditional=False) | Expected value of a function (of one argument) with respect to the distribution.
  median(n, a, b, loc=0) | Median of the distribution.
  mean(n, a, b, loc=0) | Mean of the distribution.
  var(n, a, b, loc=0) | Variance of the distribution.
  std(n, a, b, loc=0) | Standard deviation of the distribution.
  interval(confidence, n, a, b, loc=0) | Confidence interval with equal areas around the median.