Student's t-Distribution

Definition

Let $W\sim N(0,1)$ be a standard normal distribution, $V\sim {\chi }^2(r)$ be a Chi-squared Distribution, and $W,V$ be independent, then $\frac{W}{\sqrt{V/r}}\sim t(r)$ where $r\in {\mathbb{R}}^{+}$ is the degrees of freedom

Properties

PDF

$f_r(t)=\frac{\Gamma (\frac{r+1}{2})}{\sqrt{\pi r}\Gamma (\frac{r}{2})}{\left(1+\frac{t^2}{r}\right)}^{-(r+1)/2}\in \mathbb{R}$

Mean

$E(X)=0$

Variance

$\mathrm{Var}(X)=\{\begin{array}{ll}\frac{r}{r-2}& r>2\\ \infty & 1<r\le 1\end{array}$