Variance 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 variance of $X$ is given by:


 * $\var X = \dfrac k {k - 2}$

for $k > 2$, and does not exist otherwise.

Proof
By Expectation of Student's t-Distribution, we have that $\expect X$ exists $k > 2$.

Hence, take $k > 2$ from here on.

By Expectation of Student's t-Distribution, we have that for $k > 2$:


 * $\expect X = 0$

From Square of Random Variable with t-Distribution has F-Distribution, we have:


 * $\expect {X^2} = \expect Y$

with $Y \sim F_{1, k}$, where $F_{1, k}$ is the $F$-distribution with $\tuple {1, k}$ degrees of freedom.

Since $k > 2$, by Expectation of F-Distribution we have:


 * $\expect {X^2} = \dfrac k {k - 2}$

We therefore have: