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# scipy.stats.nbinom

`scipy.stats.``nbinom` = <scipy.stats._discrete_distns.nbinom_gen object>

  A negative binomial discrete random variable.

  As an instance of the `rv_discrete` class, `nbinom` 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

  Negative binomial distribution describes a sequence of i.i.d. Bernoulli trials, repeated until a predefined, non-random number of successes occurs.

  The probability mass function of the number of failures for `nbinom` is:

      nbinom.pmf(k) = choose(k+n-1, n-1) * p**n * (1-p)**k

  for `k >= 0`.

  `nbinom` takes `n` and `p` as shape parameters where n is the number of successes, whereas p is the probability of a single success.

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

  Examples

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

  Calculate a few first moments:

      >>> n, p = 0.4, 0.4
      >>> mean, var, skew, kurt = nbinom.stats(n, p, moments='mvsk')

  Display the probability mass function (`pmf`):

      >>> x = np.arange(nbinom.ppf(0.01, n, p),
      ...               nbinom.ppf(0.99, n, p))
      >>> ax.plot(x, nbinom.pmf(x, n, p), 'bo', ms=8, label='nbinom pmf')
      >>> ax.vlines(x, 0, nbinom.pmf(x, n, p), 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 = nbinom(n, p)
      >>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1,
      ...         label='frozen pmf')
      >>> ax.legend(loc='best', frameon=False)
      >>> plt.show()

  (Source code)

  Check accuracy of `cdf` and `ppf`:

      >>> prob = nbinom.cdf(x, n, p)
      >>> np.allclose(x, nbinom.ppf(prob, n, p))
      True

  Generate random numbers:

      >>> r = nbinom.rvs(n, p, size=1000)

  Methods

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