Definition 9.4.3. Student-t Distribution.
Suppose Z is a standard normal variable and Y is \(\chi^2(r)\) with Y and Z independent. Define a new random variable
\begin{equation*}
T = \frac{Z}{\sqrt{Y/r}}.
\end{equation*}
Then, T is said to have a (Student) t distribution with probability function given by
\begin{equation*}
\frac{\Gamma \left ( \frac{n+1}{2} \right ) }{\sqrt{n \pi} \; \Gamma \left ( \frac{n}{2} \right ) } \left ( 1 + \frac{x^2}{n}\right )^{ - \left ( \frac{n+1}{2} \right )}
\end{equation*}