Standard Cauchy distribution with mode = 0.

Also known as the Lorentz distribution.

Parameters :

size : int or tuple of ints

Shape of the output.

Returns :

samples : ndarray or scalar

The drawn samples.


The probability density function for the full Cauchy distribution is

P(x; x_0, \gamma) = \frac{1}{\pi \gamma \bigl[ 1+
(\frac{x-x_0}{\gamma})^2 \bigr] }

and the Standard Cauchy distribution just sets x_0=0 and \gamma=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.


..[1] NIST/SEMATECH e-Handbook of Statistical Methods, “Cauchy
..[2] Weisstein, Eric W. “Cauchy Distribution.” From MathWorld–A
Wolfram Web Resource.
..[3] Wikipedia, “Cauchy distribution”


Draw samples and plot the distribution:

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

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