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
q : array_like
a : array_like
loc : array_like, optional
scale : array_like, optional
size : int or tuple of ints, optional
moments : str, optional
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)
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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 |