scipy.stats.pearson3 = <scipy.stats._continuous_distns.pearson3_gen object>[source]#

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.


The probability density function for pearson3 is:

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


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

\(\Gamma\) 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.


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).


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

Calculate the first four moments:

>>> skew = -2
>>> 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))

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)


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.


Parameter estimates for generic data. See 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.