Student's t-distribution

Definition

Let $X$ be a standard normal random variable, $Y\sim {\chi }_n^2$ (Chi-Squared Distribution), and $X,Y$ are independent, then $T=\frac{X}{\sqrt{Y/n}}\sim t_n$, where $t_n$ is called the Student's t-distribution with $n$ degrees of freedom.

Probability density function

The PDF of Student's $t$-distribution is

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

where $n$ is the number of degrees of freedom and $\Gamma$ is the gamma function.

Properties

Mean and Variance

• Mean: $0$, since this is a symmetric distribution
• Variance: $\frac{n}{n-2}$ for $n>2$, $\infty$ for $1<n\le n$

Relation with Normal

As $n\to \infty$, $t_n\stackrel{d}{\to }\mathcal{N}(0,1)$