numpy.random.zipf(a, size=None)

Draw samples from a Zipf distribution.

Samples are drawn from a Zipf distribution with specified parameter (a), where a > 1.

The zipf distribution (also known as the zeta distribution) is a continuous probability distribution that satisfies Zipf’s law, where the frequency of an item is inversely proportional to its rank in a frequency table.


a : float

parameter, > 1.

size : {tuple, int}

Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn.


samples : {ndarray, scalar}

The returned samples are greater than or equal to one.

See also

probability density function, distribution or cumulative density function, etc.


The probability density for the Zipf distribution is

p(x) = \frac{x^{-a}}{\zeta(a)},

where \zeta is the Riemann Zeta function.

Named after the American linguist George Kingsley Zipf, who noted that the frequency of any word in a sample of a language is inversely proportional to its rank in the frequency table.


[98]Weisstein, Eric W. “Zipf Distribution.” From MathWorld–A Wolfram Web Resource.
[99]Wikipedia, “Zeta distribution”,
[100]Wikipedia, “Zipf’s Law”,
[101]Zipf, George Kingsley (1932): Selected Studies of the Principle of Relative Frequency in Language. Cambridge (Mass.).


Draw samples from the distribution:

>>> a = 2. # parameter
>>> s = np.random.zipf(a, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> import scipy.special as sps
Truncate s values at 50 so plot is interesting
>>> count, bins, ignored = plt.hist(s[s<50], 50, normed=True)
>>> x = arange(1., 50.)
>>> y = x**(-a)/sps.zetac(a)
>>> plt.plot(x, y/max(y), linewidth=2, color='r')

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