# numpy.random.Generator.standard_cauchy

method

random.Generator.standard_cauchy(size=None)

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

  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

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

  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.

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

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