F-distribution

A random variable $X$ is has an $F$ distribution with parameters $a > 0$ and $b > 0$ , written as $X \sim F (a, b)$ , if the pdf is

f (x; a, b) = {\begin{cases} 0 & x \leq 0 \\ \frac{Γ (\frac{a + b}{2})}{Γ (\frac{a}{2}) Γ (\frac{b}{2})} {(\frac{a}{b})}^{a / 2} x^{a / 2 - 1} {(1 + \frac{a}{b} x)}^{- (a + b) / 2} & x > 0 \end{cases}

We have that the cdf is

F (x; a, b) = I_{\frac{a x}{a x + b}} (\frac{a}{2}, \frac{b}{2})

We have that

Let $X \sim χ^{2} (a)$ and $Y \sim χ^{2} (b)$ be indpendent random variables. Then

\frac{X / a}{Y / b} \sim F (a, b)

We can calculate the $n$ th moment of $X$ , as

E [X^{n}] = {(\frac{b}{a})}^{n} \frac{Γ (a / 2 + n)}{Γ (a / 2)} \frac{Γ (b / 2 - n)}{Γ (b / 2)}

there’s no moment generating function.

And if $X \sim F (a, b)$ , then

\frac{1}{X} \sim F (b, a)