Distribution

Student’s t-Distribution

We say $T$ follows a Student’s t-distribution, $T\sim T(n)$, with $n$ degrees of freedom, if $T$ can be written as a ratio of two independent Random Variables, $Z$ and $\sqrt{\frac{W}{n}}$.

Therefore, $T=\frac{Z}{\sqrt{\frac{W}{n}}}$ where

• $Z\sim N(0,1)$
• $W\sim {\chi }^2(n)$

Reminder that $W=Z_1^2+\cdots +Z_n^2$ as you’ve seen in the Chi-Squared Distribution

Support of T: $(-\infty ,\infty )$ Expectation and Variance

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

PDF: $f(t)=\frac{\Gamma (\frac{n+1}{2})(1+\frac{t^2}{n}{)}^{\frac{-n+1}{2}}}{\sqrt{n\pi }\Gamma (\frac{n}{2})}$

For large $n$, we use a Z-Table, since as $n\to \infty$, $T\to Z$

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.

Related

• Chi-Squared Distribution