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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*}
The good news is that this distribution is useful and its properties are presented below without proof.

Theorem 9.4.4. Student t-distribution properties.

For the Student t variable T defined above,
\begin{equation*} \mu = 0 \end{equation*}
and if r>2
\begin{equation*} \sigma^2 = \frac{r}{r-2} \end{equation*}
and if r>3
\begin{equation*} \gamma_1 = 0 \end{equation*}
and if r>4
\begin{equation*} \gamma_2 = \frac{6}{r-4} + 3. \end{equation*}
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