# scipy.stats.pareto

scipy.stats.pareto = <scipy.stats._continuous_distns.pareto_gen object>

  A Pareto continuous random variable.

  As an instance of the `rv_continuous` class, `pareto` 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 density function for `pareto` is:

  \[f(x, b) = \frac{b}{x^{b+1}}\]

  for \(x \ge 1\), \(b > 0\).

  `pareto` takes `b` as a shape parameter for \(b\).

  The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the `loc` and `scale` parameters. Specifically, `pareto.pdf(x, b, loc, scale)` is identically equivalent to `pareto.pdf(y, b) / scale` with `y = (x - loc) / scale`. Note that shifting the location of a distribution does not make it a “noncentral” distribution; noncentral generalizations of some distributions are available in separate classes.

  Examples

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

  Calculate the first four moments:

      >>> b = 2.62
      >>> mean, var, skew, kurt = pareto.stats(b, moments='mvsk')

  Display the probability density function (`pdf`):

      >>> x = np.linspace(pareto.ppf(0.01, b),
      ...                 pareto.ppf(0.99, b), 100)
      >>> ax.plot(x, pareto.pdf(x, b),
      ...        'r-', lw=5, alpha=0.6, label='pareto pdf')

  Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed.

  Freeze the distribution and display the frozen `pdf`:

      >>> rv = pareto(b)
      >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

  Check accuracy of `cdf` and `ppf`:

      >>> vals = pareto.ppf([0.001, 0.5, 0.999], b)
      >>> np.allclose([0.001, 0.5, 0.999], pareto.cdf(vals, b))
      True

  Generate random numbers:

      >>> r = pareto.rvs(b, size=1000)

  And compare the histogram:

      >>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
      >>> ax.legend(loc='best', frameon=False)
      >>> plt.show()

  Methods

  rvs(b, loc=0, scale=1, size=1, random_state=None) | Random variates.
  pdf(x, b, loc=0, scale=1) | Probability density function.
  logpdf(x, b, loc=0, scale=1) | Log of the probability density function.
  cdf(x, b, loc=0, scale=1) | Cumulative distribution function.
  logcdf(x, b, loc=0, scale=1) | Log of the cumulative distribution function.
  sf(x, b, loc=0, scale=1) | Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
  logsf(x, b, loc=0, scale=1) | Log of the survival function.
  ppf(q, b, loc=0, scale=1) | Percent point function (inverse of `cdf` — percentiles).
  isf(q, b, loc=0, scale=1) | Inverse survival function (inverse of `sf`).
  moment(n, b, loc=0, scale=1) | Non-central moment of order n
  stats(b, loc=0, scale=1, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
  entropy(b, loc=0, scale=1) | (Differential) entropy of the RV.
  fit(data) | Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments.
  expect(func, args=(b,), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds) | Expected value of a function (of one argument) with respect to the distribution.
  median(b, loc=0, scale=1) | Median of the distribution.
  mean(b, loc=0, scale=1) | Mean of the distribution.
  var(b, loc=0, scale=1) | Variance of the distribution.
  std(b, loc=0, scale=1) | Standard deviation of the distribution.
  interval(alpha, b, loc=0, scale=1) | Endpoints of the range that contains fraction alpha [0, 1] of the distribution

scipy.stats.norminvgauss scipy.stats.pearson3