Variance of Student's t-Distribution

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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 if and only if $k > 2$.

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:

\(\ds \var X\) \(=\) \(\ds \expect {X^2} - \paren {\expect X}^2\) Variance as Expectation of Square minus Square of Expectation
\(\ds \) \(=\) \(\ds \frac k {k - 2} - 0^2\)
\(\ds \) \(=\) \(\ds \frac k {k - 2}\)

$\blacksquare$