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# scipy.stats.betaprime

scipy.stats.betaprime = <scipy.stats._continuous_distns.betaprime_gen object>

  A beta prime continuous random variable.

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

  Notes

  The probability density function for `betaprime` is:

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

  for \(x >= 0\), \(a > 0\), \(b > 0\), where \(\beta(a, b)\) is the beta function (see `scipy.special.beta`).

  `betaprime` takes `a` and `b` as shape parameters.

  The distribution is related to the `beta` distribution as follows: If \(X\) follows a beta distribution with parameters \(a, b\), then \(Y = X/(1-X)\) has a beta prime distribution with parameters \(a, b\) ([1]).

  The beta prime distribution is a reparametrized version of the F distribution. The beta prime distribution with shape parameters `a` and `b` and `scale = s` is equivalent to the F distribution with parameters `d1 = 2*a`, `d2 = 2*b` and `scale = (a/b)*s`. For example,

      >>> from scipy.stats import betaprime, f
      >>> x = [1, 2, 5, 10]
      >>> a = 12
      >>> b = 5
      >>> betaprime.pdf(x, a, b, scale=2)
      array([0.00541179, 0.08331299, 0.14669185, 0.03150079])
      >>> f.pdf(x, 2*a, 2*b, scale=(a/b)*2)
      array([0.00541179, 0.08331299, 0.14669185, 0.03150079])

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

  References

  [1]

  Beta prime distribution, Wikipedia, https://en.wikipedia.org/wiki/Beta_prime_distribution

  Examples

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

  Get the support:

      >>> a, b = 5, 6
      >>> lb, ub = betaprime.support(a, b)

  Calculate the first four moments:

      >>> mean, var, skew, kurt = betaprime.stats(a, b, moments='mvsk')

  Display the probability density function (`pdf`):

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

  Check accuracy of `cdf` and `ppf`:

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

  Generate random numbers:

      >>> r = betaprime.rvs(a, b, 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()