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 and only if it has probability density function:
- $\map {f_X} x = \dfrac 1 {\pi \lambda \paren {1 + \paren {\frac {x - \gamma} \lambda}^2} }$
for:
- $\lambda \in \R_{>0}$
- $\gamma \in \R$
This is written:
- $X \sim \Cauchy \gamma \lambda$
Scale Parameter
Let $X$ be a continuous random variable with a Cauchy distribution:
- $\map {f_X} x = \dfrac 1 {\pi \lambda \paren {1 + \paren {\frac {x - \gamma} \lambda}^2} }$
for:
- $\lambda \in \R_{>0}$
- $\gamma \in \R$
The parameter $\lambda$ is referred to as the scale parameter of $X$.
Standard
The standard Cauchy distribution is the Cauchy distribution with $\lambda = 1$ and $\gamma = 0$:
- $\map {f_X} x = \dfrac 1 {\pi \paren {1 + x^2} }$
Also presented as
The Cauchy Distribution of $X$ can be found expressed in the form:
- $\map {f_X} x = \dfrac \lambda {\pi \paren {\lambda^2 + \paren {x - \gamma}^2} }$
Also defined as
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
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $15$: Probability distributions
- Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CauchyDistribution.html