Definition:Cauchy Distribution/Also defined as

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Cauchy Distribution: Also defined as

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

Let $\Img X = \R$.

Some sources define the Cauchy distribution of $X$ as the standard Cauchy distribution:

$\map {f_X} x = \dfrac 1 {\pi \paren {1 + x^2} }$

which is obtained from the full form by setting:

$\lambda = 1$
$\gamma = 0$


Some sources give it as:

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

which is obtained from the full form by setting $\lambda = 1$.


Notation

The notation used to denote a Cauchy distribution is generally consistent, except for the parameter labels.

Some sources use $\theta$ for $\gamma$.

MathWorld uses $m$ and $b$ for $\lambda$ and $\gamma$.

All notation is perfectly good, as long as it is clear what the parameters are.


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 \(\Cauchy {\gamma} {\lambda}\) is \Cauchy {\gamma} {\lambda} .

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

\Cauchy \gamma \lambda


Sources

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