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.
Cauchy(location = 0, scale = 1)
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:
set.seed(27) X <- Cauchy(10, 0.2) X mean(X) variance(X) skewness(X) kurtosis(X) random(X, 10) pdf(X, 2) log_pdf(X, 2) cdf(X, 2) quantile(X, 0.7) cdf(X, quantile(X, 0.7)) quantile(X, cdf(X, 7))
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