scipy.stats.alpha — SciPy v1.15.3 Manual (original) (raw)

scipy.stats.alpha = <scipy.stats._continuous_distns.alpha_gen object>[source]#

An alpha continuous random variable.

As an instance of the rv_continuous class, alpha 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 alpha ([1], [2]) is:

\[f(x, a) = \frac{1}{x^2 \Phi(a) \sqrt{2\pi}} * \exp(-\frac{1}{2} (a-1/x)^2)\]

where \(\Phi\) is the normal CDF, \(x > 0\), and \(a > 0\).

alpha takes a as a shape parameter.

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, alpha.pdf(x, a, loc, scale) is identically equivalent to alpha.pdf(y, a) / scale withy = (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.

References

[1]

Johnson, Kotz, and Balakrishnan, “Continuous Univariate Distributions, Volume 1”, Second Edition, John Wiley and Sons, p. 173 (1994).

[2]

Anthony A. Salvia, “Reliability applications of the Alpha Distribution”, IEEE Transactions on Reliability, Vol. R-34, No. 3, pp. 251-252 (1985).

Examples

import numpy as np from scipy.stats import alpha import matplotlib.pyplot as plt fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

a = 3.57 mean, var, skew, kurt = alpha.stats(a, moments='mvsk')

Display the probability density function (pdf):

x = np.linspace(alpha.ppf(0.01, a), ... alpha.ppf(0.99, a), 100) ax.plot(x, alpha.pdf(x, a), ... 'r-', lw=5, alpha=0.6, label='alpha 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 = alpha(a) ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

vals = alpha.ppf([0.001, 0.5, 0.999], a) np.allclose([0.001, 0.5, 0.999], alpha.cdf(vals, a)) True

Generate random numbers:

r = alpha.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