# scipy.stats.gamma¶

scipy.stats.gamma = <scipy.stats._continuous_distns.gamma_gen object at 0x2b45d2fd2150>[source]

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=’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.

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()
``` 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

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