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# scipy.stats.burr12

scipy.stats.burr12 = <scipy.stats._continuous_distns.burr12_gen object>

  A Burr (Type XII) continuous random variable.

  As an instance of the `rv_continuous` class, `burr12` 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.

  Methods

  rvs(c, d, loc=0, scale=1, size=1, random_state=None) | Random variates.
  pdf(x, c, d, loc=0, scale=1) | Probability density function.
  logpdf(x, c, d, loc=0, scale=1) | Log of the probability density function.
  cdf(x, c, d, loc=0, scale=1) | Cumulative distribution function.
  logcdf(x, c, d, loc=0, scale=1) | Log of the cumulative distribution function.
  sf(x, c, d, loc=0, scale=1) | Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
  logsf(x, c, d, loc=0, scale=1) | Log of the survival function.
  ppf(q, c, d, loc=0, scale=1) | Percent point function (inverse of `cdf` — percentiles).
  isf(q, c, d, loc=0, scale=1) | Inverse survival function (inverse of `sf`).
  moment(order, c, d, loc=0, scale=1) | Non-central moment of the specified order.
  stats(c, d, loc=0, scale=1, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
  entropy(c, d, 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=(c, d), 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(c, d, loc=0, scale=1) | Median of the distribution.
  mean(c, d, loc=0, scale=1) | Mean of the distribution.
  var(c, d, loc=0, scale=1) | Variance of the distribution.
  std(c, d, loc=0, scale=1) | Standard deviation of the distribution.
  interval(confidence, c, d, loc=0, scale=1) | Confidence interval with equal areas around the median.

  See also

  `fisk`

    a special case of either `burr` or `burr12` with `d=1`

  `burr`

    Burr Type III distribution

  Notes

  The probability density function for `burr12` is:

  \[f(x; c, d) = c d \frac{x^{c-1}} {(1 + x^c)^{d + 1}}\]

  for \(x >= 0\) and \(c, d > 0\).

  `burr12` takes `c` and `d` as shape parameters for \(c\) and \(d\).

  This is the PDF corresponding to the twelfth CDF given in Burr’s list; specifically, it is equation (20) in Burr’s paper [1].

  The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the `loc` and `scale` parameters. Specifically, `burr12.pdf(x, c, d, loc, scale)` is identically equivalent to `burr12.pdf(y, c, d) / 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.

  The Burr type 12 distribution is also sometimes referred to as the Singh-Maddala distribution from NIST [2].

  References

  [1]

  Burr, I. W. “Cumulative frequency functions”, Annals of Mathematical Statistics, 13(2), pp 215-232 (1942).

  [2]

  https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/b12pdf.htm

  [3]

  “Burr distribution”, https://en.wikipedia.org/wiki/Burr_distribution

  Examples

      >>> import numpy as np
      >>> from scipy.stats import burr12
      >>> import matplotlib.pyplot as plt
      >>> fig, ax = plt.subplots(1, 1)

  Get the support:

      >>> c, d = 10, 4
      >>> lb, ub = burr12.support(c, d)

  Calculate the first four moments:

      >>> mean, var, skew, kurt = burr12.stats(c, d, moments='mvsk')

  Display the probability density function (`pdf`):

      >>> x = np.linspace(burr12.ppf(0.01, c, d),
      ...                 burr12.ppf(0.99, c, d), 100)
      >>> ax.plot(x, burr12.pdf(x, c, d),
      ...        'r-', lw=5, alpha=0.6, label='burr12 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 = burr12(c, d)
      >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

  Check accuracy of `cdf` and `ppf`:

      >>> vals = burr12.ppf([0.001, 0.5, 0.999], c, d)
      >>> np.allclose([0.001, 0.5, 0.999], burr12.cdf(vals, c, d))
      True

  Generate random numbers:

      >>> r = burr12.rvs(c, d, size=1000)

  And compare the histogram:

      >>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
      >>> ax.set_xlim([x[0], x[-1]])
      >>> ax.legend(loc='best', frameon=False)
      >>> plt.show()