StudentTDistribution

The Student-t distribution with $\nu$ degrees of freedom.

StudentTDistribution(nu)an inert distribution object — carries

\nu

, unevaluated. Only the single-parameter (not the location-scale 3-argument Wolfram) form is implemented — a documented scope limit.

Typed signatures

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

Domain: Statistics

Details

$PDF(x) = \dfrac{\Gamma((\nu+1)/2)}{\sqrt{\nu\pi}\,\Gamma(\nu/2)}\left(1+\dfrac{x^2}{\nu}\right)^{-(\nu+1)/2}$ , exact.
$CDF(x) = \tfrac12 I_z(\nu/2, 1/2)$ for $x<0$ , $1 - \tfrac12 I_z(\nu/2, 1/2)$ for $x\geq0$ , where $z=\nu/(\nu+x^2)$ and $I$ is BetaRegularized — exact (Abramowitz & Stegun 26.7.1).
$Mean = 0$ unconditionally — exact only for $\nu>1$ ; a documented divergence for $\nu \leq 1$ , where the mean is actually undefined.
$Variance = \nu/(\nu-2)$ unconditionally — exact only for $\nu>2$ ; same divergence policy for $\nu \leq 2$ .
RandomVariate samples $Z/\sqrt{V/\nu}$ for a standard normal $Z$ and an independent ChiSquareDistribution( $\nu$ ) draw $V$ .