# Student’s t-Distribution

We say $T$ follows a Student’s t-distribution, $T∼T(n)$, with $n$ degrees of freedom, if $T$ can be written as a ratio of two independent Random Variables, $Z$ and $nW $.

Therefore, $T=nW Z $ where

- $Z∼N(0,1)$
- $W∼χ_{2}(n)$

Reminder that $W=Z_{1}+⋯+Z_{n}$ as you’ve seen in the Chi-Squared Distribution

Support of T: $(−∞,∞)$ Expectation and Variance

- $E(T)=0$, since $T$ is symmetric around $0$
- $Var(T)=?$

PDF: $f(t)=nπ Γ(2n )Γ(2n+1 )(1+nt )_{2−n+1} $

For large $n$, we use a Z-Table, since as $n→∞$, $T→Z$

### T-Table

What’s the difference between this and Chi-Squared?

A T-Table is symmetric around it’s mean, which a Chi-Square Distribution is not.