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# scipy.stats.pearson3

scipy.stats.pearson3 = <scipy.stats._continuous_distns.pearson3_gen object>

  A pearson type III continuous random variable.

  As an instance of the `rv_continuous` class, `pearson3` 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(skew, loc=0, scale=1, size=1, random_state=None) | Random variates.
  pdf(x, skew, loc=0, scale=1) | Probability density function.
  logpdf(x, skew, loc=0, scale=1) | Log of the probability density function.
  cdf(x, skew, loc=0, scale=1) | Cumulative distribution function.
  logcdf(x, skew, loc=0, scale=1) | Log of the cumulative distribution function.
  sf(x, skew, loc=0, scale=1) | Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
  logsf(x, skew, loc=0, scale=1) | Log of the survival function.
  ppf(q, skew, loc=0, scale=1) | Percent point function (inverse of `cdf` — percentiles).
  isf(q, skew, loc=0, scale=1) | Inverse survival function (inverse of `sf`).
  moment(order, skew, loc=0, scale=1) | Non-central moment of the specified order.
  stats(skew, loc=0, scale=1, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
  entropy(skew, 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=(skew,), 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(skew, loc=0, scale=1) | Median of the distribution.
  mean(skew, loc=0, scale=1) | Mean of the distribution.
  var(skew, loc=0, scale=1) | Variance of the distribution.
  std(skew, loc=0, scale=1) | Standard deviation of the distribution.
  interval(confidence, skew, loc=0, scale=1) | Confidence interval with equal areas around the median.

  Notes

  The probability density function for `pearson3` is:

  \[f(x, \kappa) = \frac{|\beta|}{\Gamma(\alpha)} (\beta (x - \zeta))^{\alpha - 1} \exp(-\beta (x - \zeta))\]

  where:

  \[ \begin{align}\begin{aligned}\beta = \frac{2}{\kappa}\\\alpha = \beta^2 = \frac{4}{\kappa^2}\\\zeta = -\frac{\alpha}{\beta} = -\beta\end{aligned}\end{align} \]

  \(\Gamma\) is the gamma function (`scipy.special.gamma`). Pass the skew \(\kappa\) into `pearson3` as the shape parameter `skew`.

  The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the `loc` and `scale` parameters. Specifically, `pearson3.pdf(x, skew, loc, scale)` is identically equivalent to `pearson3.pdf(y, skew) / 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

  R.W. Vogel and D.E. McMartin, “Probability Plot Goodness-of-Fit and Skewness Estimation Procedures for the Pearson Type 3 Distribution”, Water Resources Research, Vol.27, 3149-3158 (1991).

  L.R. Salvosa, “Tables of Pearson’s Type III Function”, Ann. Math. Statist., Vol.1, 191-198 (1930).

  “Using Modern Computing Tools to Fit the Pearson Type III Distribution to Aviation Loads Data”, Office of Aviation Research (2003).

  Examples

      >>> import numpy as np
      >>> from scipy.stats import pearson3
      >>> import matplotlib.pyplot as plt
      >>> fig, ax = plt.subplots(1, 1)

  Get the support:

      >>> skew = -2
      >>> lb, ub = pearson3.support(skew)

  Calculate the first four moments:

      >>> mean, var, skew, kurt = pearson3.stats(skew, moments='mvsk')

  Display the probability density function (`pdf`):

      >>> x = np.linspace(pearson3.ppf(0.01, skew),
      ...                 pearson3.ppf(0.99, skew), 100)
      >>> ax.plot(x, pearson3.pdf(x, skew),
      ...        'r-', lw=5, alpha=0.6, label='pearson3 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 = pearson3(skew)
      >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

  Check accuracy of `cdf` and `ppf`:

      >>> vals = pearson3.ppf([0.001, 0.5, 0.999], skew)
      >>> np.allclose([0.001, 0.5, 0.999], pearson3.cdf(vals, skew))
      True

  Generate random numbers:

      >>> r = pearson3.rvs(skew, 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()