t Distribution

The ratio between the Standard Normal Distribution and Chi-Squared Distribution.

Definition

If $Z\sim N(0,1)$ and $V\sim {\mathcal{X}}^2(n)$ with $Z\perp V$, Then $T=\frac{Z}{\sqrt{\frac{V}{n}}}$ follows a t-distribution with degrees of freedom $n$.

Sample Mean and Variance

$$\frac{\frac{{\overline{X}}_n-\mu }{\sqrt{\frac{{\sigma }^2}{n}}}}{\sqrt{\frac{S_n^2}{{\sigma }^2}}}=\frac{{\overline{X}}_n-\mu }{\sqrt{\frac{S_n^2}{n}}}\sim t(n-1)$$

Properties

$T=\frac{Z}{\sqrt{V/n}}\sim t_{(n)}⟹T^2=\frac{Z^2}{\frac{V}{n}}\sim F(1,n)$