Student's t distribution

A Student’s $t$ Distribution is a kind of Sampling Distribution formed by combining a Z Distribution and Chi Squared Distribution in the following fashion

$$T\equiv \frac{Z}{\sqrt{{\chi }_{(k)}^2/k}}\sim t_{(k)}$$

• $t$ shows up when we try to estimate $\mu$ when the ${\sigma }^2$ is unknown.
• $t_{(k)}$ looks like $Z$ but with a fatter tail to account for the extra uncertainty of using $S^2$ instead of ${\sigma }^2$
• $t_{(k\to \infty )}=Z$
• $E[T]=0$
• $\text{Var}(T)=\frac{k}{k-2}\text{for }k>2$

Student’s t Quantiles

If $T\sim t_{(k)}$ then the $(1-\alpha )$ quantile is denoted $t_{\alpha ,k}$

• where $P(T>t_{\alpha ,k})=1-F(t_{\alpha ,k})=\alpha$
• where $F$ is the CDF
• where $\alpha$ is type I error
• where $k$ is Degree of Freedom
• where the distribution is symmetrical so $t_{1-\alpha ,k}=-t_{\alpha ,k}$