scipy.stats.gamma

scipy.stats.gamma = <scipy.stats._continuous_distns.gamma_gen object at 0x7fbe1dba4650>

A gamma continuous random variable.

Continuous random variables are defined from a standard form and may require some shape parameters to complete its specification. Any optional keyword parameters can be passed to the methods of the RV object as given below:

Parameters:

x : array_like

quantiles

q : array_like

lower or upper tail probability

a : array_like

shape parameters

loc : array_like, optional

location parameter (default=0)

scale : array_like, optional

scale parameter (default=1)

size : int or tuple of ints, optional

shape of random variates (default computed from input arguments )

moments : str, optional

composed of letters [‘mvsk’] specifying which moments to compute where ‘m’ = mean, ‘v’ = variance, ‘s’ = (Fisher’s) skew and ‘k’ = (Fisher’s) kurtosis. Default is ‘mv’.

Alternatively, the object may be called (as a function) to fix the shape,

location, and scale parameters returning a “frozen” continuous RV object:

rv = gamma(a, loc=0, scale=1)

• Frozen RV object with the same methods but holding the given shape, location, and scale fixed.

See also

erlang, expon

Notes

The probability density function for gamma is:

gamma.pdf(x, a) = lambda**a * x**(a-1) * exp(-lambda*x) / gamma(a)

for x >= 0, a > 0. Here gamma(a) refers to the gamma function.

The scale parameter is equal to scale = 1.0 / lambda.

gamma has a shape parameter a which needs to be set explicitly. For instance:

>>> from scipy.stats import gamma
>>> rv = gamma(3., loc = 0., scale = 2.)

produces a frozen form of gamma with shape a = 3., loc =0. and lambda = 1./scale = 1./2..

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

Examples

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

Calculate a few first moments:

>>> a = 1.99323054838
>>> 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, 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, normed=True, histtype='stepfilled', alpha=0.2)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()

(Source code)

Methods

rvs(a, loc=0, scale=1, size=1)	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 density function.
logcdf(x, a, loc=0, scale=1)	Log of the cumulative density function.
sf(x, a, loc=0, scale=1)	Survival function (1-cdf — 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, a, loc=0, scale=1)	Parameter estimates for generic data.
expect(func, 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 alpha percent of the distribution