Definition:Cauchy Distribution/Also presented as
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Cauchy Distribution: Also presented as
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $\Img X = \R$.
The Cauchy Distribution of $X$ can be found expressed in the form:
- $\map {f_X} x = \dfrac \lambda {\pi \paren {\lambda^2 + \paren {x - \gamma}^2} }$
for:
- $\lambda \in \R_{>0}$
- $\gamma \in \R$
This is written:
- $X \sim \Cauchy \gamma \lambda$
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
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cauchy distribution
- Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CauchyDistribution.html