scipy.stats.pareto

scipy.stats.pareto = <scipy.stats._continuous_distns.pareto_gen object>

A Pareto continuous random variable.

As an instance of the rv_continuous class, pareto 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 pareto is:

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

for \(x \ge 1\), \(b > 0\).

pareto takes \(b\) 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, pareto.pdf(x, b, loc, scale) is identically equivalent to pareto.pdf(y, b) / scale with y = (x - loc) / scale.

Examples

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

Calculate a few first moments:

>>> b = 2.62
>>> mean, var, skew, kurt = pareto.stats(b, moments='mvsk')

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

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

Generate random numbers:

>>> r = pareto.rvs(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()

Methods

rvs(b, loc=0, scale=1, size=1, random_state=None)	Random variates.
pdf(x, b, loc=0, scale=1)	Probability density function.
logpdf(x, b, loc=0, scale=1)	Log of the probability density function.
cdf(x, b, loc=0, scale=1)	Cumulative distribution function.
logcdf(x, b, loc=0, scale=1)	Log of the cumulative distribution function.
sf(x, b, loc=0, scale=1)	Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).
logsf(x, b, loc=0, scale=1)	Log of the survival function.
ppf(q, b, loc=0, scale=1)	Percent point function (inverse of cdf — percentiles).
isf(q, b, loc=0, scale=1)	Inverse survival function (inverse of sf).
moment(n, b, loc=0, scale=1)	Non-central moment of order n
stats(b, loc=0, scale=1, moments=’mv’)	Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(b, loc=0, scale=1)	(Differential) entropy of the RV.
fit(data, b, loc=0, scale=1)	Parameter estimates for generic data.
expect(func, args=(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(b, loc=0, scale=1)	Median of the distribution.
mean(b, loc=0, scale=1)	Mean of the distribution.
var(b, loc=0, scale=1)	Variance of the distribution.
std(b, loc=0, scale=1)	Standard deviation of the distribution.
interval(alpha, b, loc=0, scale=1)	Endpoints of the range that contains alpha percent of the distribution