# scipy.stats.alpha¶

scipy.stats.alpha(*args, **kwds) = <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 (, ) 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 with y = (x - loc) / scale.

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

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


Calculate a few first 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, histtype='stepfilled', alpha=0.2)
>>> 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(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) 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(alpha, a, loc=0, scale=1) Endpoints of the range that contains alpha percent of the distribution

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