Variance of Student's t-Distribution/Proof 1
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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 for $k > 1$:
- $\expect X = 0$
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.
By Expectation of F-Distribution we have that $\expect Y$ exists if and only if $k > 2$.
- $\expect Y = \expect {X^2} = \dfrac k {k - 2}$
We therefore have:
\(\ds \var X\) | \(=\) | \(\ds \expect {X^2} - \paren {\expect X}^2\) | Variance as Expectation of Square minus Square of Expectation | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac k {k - 2} - 0^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac k {k - 2}\) |
$\blacksquare$