Cauchy

The Cauchy cumulative distribution function is defined for all real numbers α, and for positive β, by the integral

$\displaystyle \func$CauchyDist(x;α, β) = $\displaystyle {\frac{{1}}{{\pi \beta }}}$$\displaystyle \int_{{-\infty }}^{{x}}$$\displaystyle \left(\vphantom{ 1+\left( \frac{u-\alpha }{\beta }\right) ^{2}}\right.$1 + $\displaystyle \left(\vphantom{ \frac{u-\alpha }{\beta }}\right.$$\displaystyle {\frac{{u-\alpha }}{{\beta }}}$$\displaystyle \left.\vphantom{ \frac{u-\alpha }{\beta }}\right)^{{2}}_{}$$\displaystyle \left.\vphantom{ 1+\left( \frac{u-\alpha }{\beta }\right) ^{2}}\right)^{{-1}}_{}$du

of the Cauchy probability density function

$\displaystyle \limfunc$CauchyDen(u;α, β) = $\displaystyle {\frac{{1}}{{\pi \beta \left( 1+\left(
\frac{u-\alpha }{\beta }\right) ^{2}\right) }}}$

The median of this distribution is α. The Cauchy probability density function is symmetric about α and has a unique maximum at α.

The following plots show probability density functions $\func$CauchyDen(x;α, β) and cumulative distribution functions $\func$CauchyDist(x;α, β), for the parameters $\left(\vphantom{ \alpha ,\beta }\right.$α, β$\left.\vphantom{ \alpha ,\beta }\right)$ = (- 3, 1), (0, 1.5), and (3, 1), and -5≤x≤5.

itbpFU2.1577in1.7573in0inCauchy density functions itbpFU2.1577in1.7573in0inCauchy distribution functions