Definition:Cauchy Distribution

Definition

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $\Img X = \R$.

$X$ is said to have a Cauchy distribution if it has probability density function:

$\map {f_X} x = \dfrac 1 {\pi \gamma \paren {1 + \paren {\frac {x - x_0} \gamma}^2} }$

for some $\gamma > 0$.

This is written:

$X \sim \Cauchy {x_0} \gamma$

Also see

• Results about the Cauchy distribution can be found here.

Source of Name

This entry was named for Augustin Louis Cauchy.

Technical Note

The $\LaTeX$ code for \(\Cauchy {x_0} {\gamma}\) is \Cauchy {x_0} {\gamma} .

When either of the arguments is a single character, it is usual to omit the braces:

\Cauchy {x_0} \gamma

Sources

• Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CauchyDistribution.html