# scipy.stats.gamma¶

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, with scale = 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 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.

Examples

>>> 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, histtype='stepfilled', alpha=0.2)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show() Methods

 rvs(a, loc=0, scale=1, size=1, random_state=None) Random variates. pdf(x, a, loc=0, scale=1) Probability density function. logpdf(x, a, loc=0, scale=1) Log of the probability density function. cdf(x, a, loc=0, scale=1) Cumulative distribution function. logcdf(x, a, loc=0, scale=1) Log of the cumulative distribution function. sf(x, a, loc=0, scale=1) Survival function (also defined as 1 - cdf, but sf is sometimes more accurate). logsf(x, a, loc=0, scale=1) Log of the survival function. ppf(q, a, loc=0, scale=1) Percent point function (inverse of cdf — percentiles). isf(q, a, loc=0, scale=1) Inverse survival function (inverse of sf). moment(n, a, loc=0, scale=1) Non-central moment of order n stats(a, loc=0, scale=1, moments=’mv’) Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). entropy(a, 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,), 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, loc=0, scale=1) Median of the distribution. mean(a, loc=0, scale=1) Mean of the distribution. var(a, loc=0, scale=1) Variance of the distribution. std(a, loc=0, scale=1) Standard deviation of the distribution. interval(alpha, a, loc=0, scale=1) Endpoints of the range that contains fraction alpha [0, 1] of the distribution