Square of Random Variable with t-Distribution has F-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:
- $X^2 \sim F_{1, k}$
where $F_{1, k}$ is the $F$-distribution with $\tuple {1, k}$ degrees of freedom.
Proof
Let $Y \sim F_{1, k}$.
We aim to show that:
- $\map \Pr {Y < x^2} = \map \Pr {|X| < x}$
for all $x \ge 0$.
That is:
- $\map \Pr {Y < x^2} = \map \Pr {-x < X < x}$
for all $x \ge 0$.
We have:
\(\ds \map \Pr {Y < x^2}\) | \(=\) | \(\ds \int_0^{x^2} \frac {k^{k/2} 1^{1/2} u^{\paren {1/2} - 1} } {\paren {k + u}^{\paren {1 + k}/2} \map \Beta {1/2, k/2} } \rd u\) | Definition of F-Distribution | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {k^{k/2} } {k^{k/2} \sqrt k \map \Beta {1/2, k/2} } \int_0^{x^2} \frac 1 {\sqrt u} \paren {1 + \frac u k}^{-\paren {1 + k}/2} \rd u\) | extracting a factor of $k^{-\paren {1 + k}/2}$, rewriting | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\map \Gamma {\frac {k + 1} 2} } {\sqrt k \map \Gamma {\frac 1 2} \map \Gamma {\frac k 2} } \int_0^{x^2} \frac 1 {\sqrt u} \paren {1 + \frac u k}^{-\paren {1 + k}/2} \rd u\) | Definition of Beta Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\map \Gamma {\frac {k + 1} 2} } {\sqrt {\pi k} \map \Gamma {\frac k 2} } \int_0^{x^2} \frac 1 {\sqrt u} \paren {1 + \frac u k}^{-\paren {1 + k}/2} \rd u\) | Gamma Function of One Half | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\map \Gamma {\frac {k + 1} 2} } {\sqrt {\pi k} \map \Gamma {\frac k 2} } \int_0^x \frac {2 v} {\sqrt {v^2} } \paren {1 + \frac {v^2} k}^{-\paren {1 + k}/2} \rd v\) | substituting $u = v^2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\map \Gamma {\frac {k + 1} 2} } {\sqrt {\pi k} \map \Gamma {\frac k 2} } \int_0^x 2 \paren {1 + \frac {v^2} k}^{-\paren {1 + k}/2} \rd v\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\map \Gamma {\frac {k + 1} 2} } {\sqrt {\pi k} \map \Gamma {\frac k 2} } \int_{-x}^x \paren {1 + \frac {v^2} k}^{-\paren {1 + k}/2} \rd v\) | Definite Integral of Even Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Pr {-x < X < x}\) | Definition of Student's t-Distribution |
$\blacksquare$