Excess Kurtosis of Student's t-Distribution

Theorem
Let $k$ be a strictly positive integer.

Let $X \sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom.

Then the excess kurtosis $\gamma_2$ of $X$ is given by:


 * $\gamma_2 = \dfrac 6 {k - 4}$

for $k > 4$, and does not exist otherwise.