random.Generator.pareto(a, size=None)#

Draw samples from a Pareto II (AKA Lomax) distribution with specified shape.

afloat or array_like of floats

Shape of the distribution. Must be positive.

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 Pareto II distribution.

See also


Pareto I distribution


Lomax (Pareto II) distribution


Generalized Pareto distribution


The probability density for the Pareto II distribution is

\[p(x) = \frac{a}{{x+1}^{a+1}} , x \ge 0\]

where \(a > 0\) is the shape.

The Pareto II distribution is a shifted and scaled version of the Pareto I distribution, which can be found in scipy.stats.pareto.



Francis Hunt and Paul Johnson, On the Pareto Distribution of Sourceforge projects.


Pareto, V. (1896). Course of Political Economy. Lausanne.


Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme Values, Birkhauser Verlag, Basel, pp 23-30.


Wikipedia, “Pareto distribution”,


Draw samples from the distribution:

>>> a = 3.
>>> rng = np.random.default_rng()
>>> s = rng.pareto(a, 10000)

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

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 3, 50)
>>> pdf = a / (x+1)**(a+1)
>>> plt.hist(s, bins=x, density=True, label='histogram')
>>> plt.plot(x, pdf, linewidth=2, color='r', label='pdf')
>>> plt.xlim(x.min(), x.max())
>>> plt.legend()