`numpy.random._generator:Generator.standard_cauchy`

Signature

def standard_cauchy ( size = None )

Summary

Draw samples from a standard Cauchy distribution with mode = 0.

Extended Summary

Also known as the Lorentz distribution.

Parameters

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. Default is None, in which case a single value is returned.

Returns

samples : ndarray or scalar: The drawn samples.

Notes

The probability density function for the full Cauchy distribution is

P (x; x_{0}, γ) = \frac{1}{π γ [ 1 + ( \frac{x - x _{0}}{γ} ) ^{2} ]}

and the Standard Cauchy distribution just sets $x_{0} = 0$ and $γ = 1$

The Cauchy distribution arises in the solution to the driven harmonic oscillator problem, and also describes spectral line broadening. It also describes the distribution of values at which a line tilted at a random angle will cut the x axis.

When studying hypothesis tests that assume normality, seeing how the tests perform on data from a Cauchy distribution is a good indicator of their sensitivity to a heavy-tailed distribution, since the Cauchy looks very much like a Gaussian distribution, but with heavier tails.

Examples

Draw samples and plot the distribution:

import matplotlib.pyplot as plt
rng = np.random.default_rng()
s = rng.standard_cauchy(1000000)
s = s[(s>-25) & (s<25)]  # truncate distribution so it plots well

✓

plt.hist(s, bins=100)

✗

plt.show()

✓

Aliases

numpy.random.Generator.standard_cauchy