Draw samples from a standard Cauchy distribution with mode = 0.
Also known as the Lorentz distribution.
- sizeint or tuple of ints, optional
Output shape. If the given shape is, e.g.,
(m, n, k), then
m * n * ksamples are drawn. Default is None, in which case a single value is returned.
- samplesndarray or scalar
The drawn samples.
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.
NIST/SEMATECH e-Handbook of Statistical Methods, “Cauchy Distribution”, https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm
Weisstein, Eric W. “Cauchy Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/CauchyDistribution.html
Wikipedia, “Cauchy distribution” https://en.wikipedia.org/wiki/Cauchy_distribution
Draw samples and plot the distribution:
>>> import matplotlib.pyplot as plt >>> s = np.random.default_rng().standard_cauchy(1000000) >>> s = s[(s>-25) & (s<25)] # truncate distribution so it plots well >>> plt.hist(s, bins=100) >>> plt.show()