beta

Python_modules.mmcomplexity.beta = <scipy.stats._continuous_distns.beta_gen object>

A beta continuous random variable.

As an instance of the rv_continuous class, beta 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.

rvs(a, b, loc=0, scale=1, size=1, random_state=None)

Random variates.

pdf(x, a, b, loc=0, scale=1)

Probability density function.

logpdf(x, a, b, loc=0, scale=1)

Log of the probability density function.

cdf(x, a, b, loc=0, scale=1)

Cumulative distribution function.

logcdf(x, a, b, loc=0, scale=1)

Log of the cumulative distribution function.

sf(x, a, b, loc=0, scale=1)

Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).

logsf(x, a, b, loc=0, scale=1)

Log of the survival function.

ppf(q, a, b, loc=0, scale=1)

Percent point function (inverse of cdf — percentiles).

isf(q, a, b, loc=0, scale=1)

Inverse survival function (inverse of sf).

moment(n, a, b, loc=0, scale=1)

Non-central moment of order n

stats(a, b, loc=0, scale=1, moments=’mv’)

Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).

entropy(a, b, loc=0, scale=1)

(Differential) entropy of the RV.

fit(data, a, b, loc=0, scale=1)

Parameter estimates for generic data.

expect(func, args=(a, b), 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, b, loc=0, scale=1)

Median of the distribution.

mean(a, b, loc=0, scale=1)

Mean of the distribution.

var(a, b, loc=0, scale=1)

Variance of the distribution.

std(a, b, loc=0, scale=1)

Standard deviation of the distribution.

interval(alpha, a, b, loc=0, scale=1)

Endpoints of the range that contains alpha percent of the distribution

The probability density function for beta is:

\[f(x, a, b) = \frac{\Gamma(a+b) x^{a-1} (1-x)^{b-1}} {\Gamma(a) \Gamma(b)}\]

for \(0 < x < 1\), \(a > 0\), \(b > 0\), where \(\Gamma\) is the gamma function (scipy.special.gamma).

beta takes \(a\) and \(b\) as shape parameters.

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

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

Calculate a few first moments:

>>> a, b = 2.31, 0.627
>>> mean, var, skew, kurt = beta.stats(a, b, moments='mvsk')

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

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

Generate random numbers:

>>> r = beta.rvs(a, b, size=1000)

And compare the histogram:

>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()