===== Student's t-Distribution =====
==== Function ====
| @#FF9944: definition  | @#FF9944: $f:$ ?? |
| @#FF9944: definition  | @#FF9944: $f(t, \nu) := \dfrac{1}{\nu^{\frac{1}{2}}{\mathrm B}(\frac{1}{2},\frac{\nu}{2})}\cdot\left(1+\dfrac{t^2}{\nu}\right)^{-\frac{\nu+1}{2}}$ |

>todo

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=== 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:

<code>
Plot[Sqrt[n/(2*Pi)]*Beta[1/2,n/2],{n,-6,6}]
</code>

>todo
>https://en.wikipedia.org/wiki/Generalized_Pareto_distribution#Characterization

== Relation to the Cauchy distribution ==
>todo

=== Theorems ===
>todo

=== Reference ===
Wikipedia: 
[[https://en.wikipedia.org/wiki/Student%27s_t-distribution|Student's t-distribution]]

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=== Requirements ===
[[Euler beta function]]
=== Related ===
[[Normal distribution]]