Student's t-Distribution

probability-theory

Definition

Student's t-Distribution

Student’s t-Distribution with $n$ degrees of freedom occurs as a scaled ratio of a standard normal and the root of a chi-squared distribution.

Let $U\sim \mathcal{N}(0,1)$ and $V\sim {\chi }_n^2$ independent, then

$$X=\frac{U}{\sqrt{\frac{V}{n}}}\sim t_n$$

Density Probability Function

The density $f$ of a $t_n$ random variable is given by:

$$f(x)=\frac{\Gamma \left(\frac{n+1}{2}\right)}{\sqrt{\pi n}\cdot \Gamma \left(\frac{n}{2}\right)}{\left(1+\frac{x^2}{n}\right)}^{-(n+1)/2}$$

for $x\in \mathbb{R}$ and $\Gamma$ being the gamma function.