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

Draw samples from a Zipf distribution.

Samples are drawn from a Zipf distribution with specified parameter a > 1.

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


a : float or array_like of floats

Distribution parameter. Should be greater than 1.

size : int or tuple of ints, optional

Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.


out : ndarray or scalar

Drawn samples from the parameterized Zipf distribution.

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.

It is named for 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.


[R283]Zipf, G. K., “Selected Studies of the Principle of Relative Frequency in Language,” Cambridge, MA: Harvard Univ. Press, 1932.


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
>>> from scipy import special

Truncate s values at 50 so plot is interesting:

>>> count, bins, ignored = plt.hist(s[s<50], 50, normed=True)
>>> x = np.arange(1., 50.)
>>> y = x**(-a) / special.zetac(a)
>>> plt.plot(x, y/max(y), linewidth=2, color='r')