scipy.stats.gamma — SciPy v1.15.3 Manual (original) (raw)

scipy.stats.gamma = <scipy.stats._continuous_distns.gamma_gen object>[source]#

A gamma continuous random variable.

As an instance of the rv_continuous class, gamma 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 gamma is:

\[f(x, a) = \frac{x^{a-1} e^{-x}}{\Gamma(a)}\]

for \(x \ge 0\), \(a > 0\). Here \(\Gamma(a)\) refers to the gamma function.

gamma takes a as a shape parameter for \(a\).

When \(a\) is an integer, gamma reduces to the Erlang distribution, and when \(a=1\) to the exponential distribution.

Gamma distributions are sometimes parameterized with two variables, with a probability density function of:

\[f(x, \alpha, \beta) = \frac{\beta^\alpha x^{\alpha - 1} e^{-\beta x }}{\Gamma(\alpha)}\]

Note that this parameterization is equivalent to the above, withscale = 1 / beta.

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

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

Calculate the first four moments:

a = 1.99 mean, var, skew, kurt = gamma.stats(a, moments='mvsk')

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

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

Generate random numbers:

r = gamma.rvs(a, 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()

Methods