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# scipy.stats.zipf

`scipy.stats.``zipf` = <scipy.stats._discrete_distns.zipf_gen object at 0x450c064c>

  A Zipf discrete random variable.

  As an instance of the `rv_discrete` class, `zipf` 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 `zipf` is:

      zipf.pmf(k, a) = 1/(zeta(a) * k**a)

  for `k >= 1`.

  `zipf` takes `a` as shape parameter.

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

  Examples

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

  Calculate a few first moments:

      >>> a = 6.5
      >>> mean, var, skew, kurt = zipf.stats(a, moments='mvsk')

  Display the probability mass function (`pmf`):

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

  Generate random numbers:

      >>> r = zipf.rvs(a, size=1000)

  Methods

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