scipy.stats.pareto

scipy.stats.pareto = <scipy.stats._continuous_distns.pareto_gen object at 0x2b45d3011690>

A Pareto 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

quantiles

q : array_like

lower or upper tail probability

b : array_like

shape parameters

loc : array_like, optional

location parameter (default=0)

scale : array_like, optional

scale parameter (default=1)

size : int or tuple of ints, optional

shape of random variates (default computed from input arguments )

moments : str, optional

composed of letters [‘mvsk’] specifying which moments to compute where ‘m’ = mean, ‘v’ = variance, ‘s’ = (Fisher’s) skew and ‘k’ = (Fisher’s) kurtosis. (default=’mv’)

Alternatively, the object may be called (as a function) to fix the shape,

location, and scale parameters returning a “frozen” continuous RV object:

rv = pareto(b, loc=0, scale=1)

• Frozen RV object with the same methods but holding the given shape, location, and scale fixed.

Notes

The probability density function for pareto is:

pareto.pdf(x, b) = b / x**(b+1)

for x >= 1, b > 0.

Examples

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

Calculate a few first moments:

>>> b = 2.62171653214
>>> 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, 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, normed=True, histtype='stepfilled', alpha=0.2)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()

(Source code)

Methods

rvs(b, loc=0, scale=1, size=1)	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 density function.
logcdf(x, b, loc=0, scale=1)	Log of the cumulative density function.
sf(x, b, loc=0, scale=1)	Survival function (1-cdf — 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, 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