scipy.stats.zipf

scipy.stats.zipf = <scipy.stats._discrete_distns.zipf_gen object at 0x7fbe1dab5650>

A Zipf discrete random variable.

Discrete 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

a : array_like

shape parameters

loc : array_like, optional

location parameter (default=0)

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 is ‘mv’.

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

location parameters returning a “frozen” discrete RV object:

rv = zipf(a, loc=0)

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

Notes

The probability mass function for zipf is:

zipf.pmf(k, a) = 1/(zeta(a) * k**a)

for k >= 1.

zipf takes a as shape parameter.

Examples

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

Calculate a few first moments:

>>> a = 6.5
>>> mean, var, skew, kurt = zipf.stats(a, moments='mvsk')

Display the probability mass function (pmf):

>>> x = np.arange(zipf.ppf(0.01, a),
...               zipf.ppf(0.99, a))
>>> ax.plot(x, zipf.pmf(x, a), 'bo', ms=8, label='zipf pmf')
>>> ax.vlines(x, 0, zipf.pmf(x, a), colors='b', lw=5, alpha=0.5)

Alternatively, freeze the distribution and display the frozen pmf:

>>> rv = zipf(a)
>>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1,
...         label='frozen pmf')
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()

(Source code)

Check accuracy of cdf and ppf:

>>> prob = zipf.cdf(x, a)
>>> np.allclose(x, zipf.ppf(prob, a))
True

Generate random numbers:

>>> r = zipf.rvs(a, size=1000)

Methods

rvs(a, loc=0, size=1)	Random variates.
pmf(x, a, loc=0)	Probability mass function.
logpmf(x, a, loc=0)	Log of the probability mass function.
cdf(x, a, loc=0)	Cumulative density function.
logcdf(x, a, loc=0)	Log of the cumulative density function.
sf(x, a, loc=0)	Survival function (1-cdf — sometimes more accurate).
logsf(x, a, loc=0)	Log of the survival function.
ppf(q, a, loc=0)	Percent point function (inverse of cdf — percentiles).
isf(q, a, loc=0)	Inverse survival function (inverse of sf).
stats(a, loc=0, moments='mv')	Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(a, loc=0)	(Differential) entropy of the RV.
expect(func, a, loc=0, lb=None, ub=None, conditional=False)	Expected value of a function (of one argument) with respect to the distribution.
median(a, loc=0)	Median of the distribution.
mean(a, loc=0)	Mean of the distribution.
var(a, loc=0)	Variance of the distribution.
std(a, loc=0)	Standard deviation of the distribution.
interval(alpha, a, loc=0)	Endpoints of the range that contains alpha percent of the distribution