Note that the Cauchy distribution is the student's t distribution with one degree of freedom. The Cauchy distribution does not have a well defined mean or variance. Cauchy distributions often appear as priors in Bayesian contexts due to their heavy tails.
The location parameter. Can be any real number. Defaults
The scale parameter. Must be greater than zero (?). Defaults
We recommend reading this documentation on https://alexpghayes.github.io/distributions3, where the math will render with additional detail and much greater clarity.
In the following, let X be a Cauchy variable with mean
location = x_0 and
scale = γ.
Support: R, the set of all real numbers
Probability density function (p.d.f):
f(x) = 1 / (π γ (1 + ((x - x_0) / γ)^2)
Cumulative distribution function (c.d.f):
F(t) = arctan((t - x_0) / γ) / π + 1/2
Moment generating function (m.g.f):
Does not exist.
Other continuous distributions:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
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