Scale Parameter of Cauchy Distribution is Half-Width at Half-Height

Definition

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:

$\lambda \in \R_{>0}$

$\gamma \in \R$

Then $\lambda$ can be referred to as the half-width at half-height:

$\map \Pr {\gamma - \lambda < X \le \gamma + \lambda} = \dfrac 1 2$

Proof

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Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cauchy distribution