ChiDistribution

The chi distribution with $k$ degrees of freedom — the distribution of $\sqrt{\chi^2_k}$ .

ChiDistribution(k)an inert distribution object — carries

k

, unevaluated.

Typed signatures

Type	Package	Applies
`(real<0..>) -> distribution`	`statistics`	always

Domain: Statistics

Details

$PDF(x) = 2^{1-k/2}x^{k-1}e^{-x^2/2}/\Gamma(k/2)$ for $x \geq 0$ .
$CDF(x) = P(k/2, x^2/2)$ via GammaRegularized (same convention as ChiSquareDistribution), clamped to $0$ below $x=0$ .
$Mean = \sqrt{2}\,\Gamma((k+1)/2)/\Gamma(k/2)$ , $Variance = k - Mean^2$ , both exact.
RandomVariate samples $\sqrt{V}$ for an independent ChiSquareDistribution( $k$ ) draw $V$ .

Examples

Basic #

\operatorname{PDF}(\mathrm{ChiDistribution}(3), 1)

Out

\frac{\sqrt{2}}{2Γ(\frac{3}{2})\sqrt{e}}

\operatorname{CDF}(\mathrm{ChiDistribution}(3), 1)

Out

1-\operatorname{GammaRegularized}(\frac{3}{2}, \frac{1}{2})

\mathrm{mean}(\mathrm{ChiDistribution}(3))

Out

\frac{\sqrt{2}Γ(2)}{Γ(\frac{3}{2})}

About

Examples	3