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; see random-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.
Notes
The probability density function for the full Cauchy distribution is
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and the Standard Cauchy distribution just sets
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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()