numpy.random.RandomState.standard_cauchy

method

RandomState.standard_cauchy(size=None)

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

Also known as the Lorentz distribution.

Note

New code should use the standard_cauchy method of a default_rng() instance instead; please see the Quick Start.

Parameters

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

Returns

samplesndarray or scalar

The drawn samples.

See also

Generator.standard_cauchy

which should be used for new code.

Notes

The probability density function for the full Cauchy distribution is

and the Standard Cauchy distribution just sets and

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.

References

1

NIST/SEMATECH e-Handbook of Statistical Methods, “Cauchy Distribution”, https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm

2

Weisstein, Eric W. “Cauchy Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/CauchyDistribution.html

3

Wikipedia, “Cauchy distribution” https://en.wikipedia.org/wiki/Cauchy_distribution

Examples

Draw samples and plot the distribution:

>>> import matplotlib.pyplot as plt
>>> s = np.random.standard_cauchy(1000000)
>>> s = s[(s>-25) & (s<25)]  # truncate distribution so it plots well
>>> plt.hist(s, bins=100)
>>> plt.show()