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 discrete probability distribution that satisfies Zipf’s law: the frequency of an item is inversely proportional to its rank in a frequency table.


New code should use the zipf method of a Generator instance instead; please see the Quick start.

afloat or array_like of floats

Distribution parameter. Must be greater than 1.

sizeint 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.

outndarray or scalar

Drawn samples from the parameterized Zipf distribution.

See also


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


which should be used for new code.


The probability density for the Zipf distribution is

\[p(k) = \frac{k^{-a}}{\zeta(a)},\]

for integers \(k \geq 1\), 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.



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 = 4.0
>>> n = 20000
>>> s = np.random.zipf(a, n)

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

>>> import matplotlib.pyplot as plt
>>> from scipy.special import zeta  

bincount provides a fast histogram for small integers.

>>> count = np.bincount(s)
>>> k = np.arange(1, s.max() + 1)
>>>, count[1:], alpha=0.5, label='sample count')
>>> plt.plot(k, n*(k**-a)/zeta(a), 'k.-', alpha=0.5,
...          label='expected count')   
>>> plt.semilogy()
>>> plt.grid(alpha=0.4)
>>> plt.legend()
>>> plt.title(f'Zipf sample, a={a}, size={n}')