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_continuousclass,- gammaobject 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 - gammais:\[f(x, a) = \frac{x^{a-1} e^{-x}}{\Gamma(a)}\]- for \(x \ge 0\), \(a > 0\). Here \(\Gamma(a)\) refers to the gamma function. - gammatakes- aas a shape parameter for \(a\).- When \(a\) is an integer, - gammareduces 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 - locand- scaleparameters. Specifically,- gamma.pdf(x, a, loc, scale)is identically equivalent to- gamma.pdf(y, a) / scalewith- 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 - >>> 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 - cdfand- 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 - 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(order, a, loc=0, scale=1) - Non-central moment of the specified order. - 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(confidence, a, loc=0, scale=1) - Confidence interval with equal areas around the median.