Student's t-Distribution

Function

definition $f:$ ??
definition $f(t, \nu) := \dfrac{1}{\nu^{\frac{1}{2}}{\mathrm B}(\frac{1}{2},\frac{\nu}{2})}\cdot\left(1+\dfrac{t^2}{\nu}\right)^{-\frac{\nu+1}{2}}$
todo

Discussion

Relation to the Normal distribution

The limit is clear using

$\lim_{n\to\infty}\left(1+\dfrac{x}{n}\right)^{a\,n+b}=\lim_{n\to\infty}\left(1+\dfrac{a\,x}{n}\right)^n={\mathrm e}^{a\,x}$.

Here the normalization:

Plot[Sqrt[n/(2*Pi)]*Beta[1/2,n/2],{n,-6,6}]
Relation to the Cauchy distribution
todo

Theorems

todo

Reference

Wikipedia: Student's t-distribution


Requirements

Euler beta function

Normal distribution