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# scipy.stats.skellam

`scipy.stats.``skellam` = <scipy.stats._discrete_distns.skellam_gen object at 0x450c094c>

  A Skellam discrete random variable.

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

  Probability distribution of the difference of two correlated or uncorrelated Poisson random variables.

  Let k1 and k2 be two Poisson-distributed r.v. with expected values lam1 and lam2. Then, `k1 - k2` follows a Skellam distribution with parameters `mu1 = lam1 - rho*sqrt(lam1*lam2)` and `mu2 = lam2 - rho*sqrt(lam1*lam2)`, where rho is the correlation coefficient between k1 and k2. If the two Poisson-distributed r.v. are independent then `rho = 0`.

  Parameters mu1 and mu2 must be strictly positive.

  For details see: http://en.wikipedia.org/wiki/Skellam_distribution

  `skellam` takes `mu1` and `mu2` as shape parameters.

  The probability mass function above is defined in the “standardized” form. To shift distribution use the `loc` parameter. Specifically, `skellam.pmf(k, mu1, mu2, loc)` is identically equivalent to `skellam.pmf(k - loc, mu1, mu2)`.

  Examples

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

  Calculate a few first moments:

      >>> mu1, mu2 = 15, 8
      >>> mean, var, skew, kurt = skellam.stats(mu1, mu2, moments='mvsk')

  Display the probability mass function (`pmf`):

      >>> x = np.arange(skellam.ppf(0.01, mu1, mu2),
      ...               skellam.ppf(0.99, mu1, mu2))
      >>> ax.plot(x, skellam.pmf(x, mu1, mu2), 'bo', ms=8, label='skellam pmf')
      >>> ax.vlines(x, 0, skellam.pmf(x, mu1, mu2), 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 = skellam(mu1, mu2)
      >>> 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 = skellam.cdf(x, mu1, mu2)
      >>> np.allclose(x, skellam.ppf(prob, mu1, mu2))
      True

  Generate random numbers:

      >>> r = skellam.rvs(mu1, mu2, size=1000)

  Methods

  `rvs(mu1, mu2, loc=0, size=1, random_state=None)` | Random variates.
  `pmf(x, mu1, mu2, loc=0)` | Probability mass function.
  `logpmf(x, mu1, mu2, loc=0)` | Log of the probability mass function.
  `cdf(x, mu1, mu2, loc=0)` | Cumulative density function.
  `logcdf(x, mu1, mu2, loc=0)` | Log of the cumulative density function.
  `sf(x, mu1, mu2, loc=0)` | Survival function (`1 - cdf` — sometimes more accurate).
  `logsf(x, mu1, mu2, loc=0)` | Log of the survival function.
  `ppf(q, mu1, mu2, loc=0)` | Percent point function (inverse of `cdf` — percentiles).
  `isf(q, mu1, mu2, loc=0)` | Inverse survival function (inverse of `sf`).
  `stats(mu1, mu2, loc=0, moments='mv')` | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
  `entropy(mu1, mu2, loc=0)` | (Differential) entropy of the RV.
  `expect(func, mu1, mu2, loc=0, lb=None, ub=None, conditional=False)` | Expected value of a function (of one argument) with respect to the distribution.
  `median(mu1, mu2, loc=0)` | Median of the distribution.
  `mean(mu1, mu2, loc=0)` | Mean of the distribution.
  `var(mu1, mu2, loc=0)` | Variance of the distribution.
  `std(mu1, mu2, loc=0)` | Standard deviation of the distribution.
  `interval(alpha, mu1, mu2, loc=0)` | Endpoints of the range that contains alpha percent of the distribution