23.3 - F Distribution

We describe a very useful distribution in Statistics known as the F distribution.

Let \(U\) and \(V\) be independent chi-square variables with \(r_1\) and \(r_2\) degrees of freedom, respectively. The joint pdf is

\(\begin{align*}
g(u, v) = \frac{ u^{r_1/2-1}e^{-u/2} v^{r_2/2-1}e^{-v/2} } { \Gamma (r_1/2) 2^{r_1/2} \Gamma
(r_2/2) 2^{r_2/2} } , \hspace{1cm} 0<u<\infty, 0<v<\infty
\end{align*}\)

Define the random variable \(W = \frac{U/r_1}{V/r_2}\)

This time we use the distribution function technique described in lesson 22,

\(\begin{align*}
F(w) = P(W \leq w)
= P \left( \frac{U/r_1}{V/r_2} \leq w \right) = P(U \leq \frac{r_1}{r_2} wV) = \int_0^\infty \int_0^{(r_1/r_2)wv} g
(u, v) du dv
\end{align*}\)

\(\begin{align*}
F(w) =\frac{1}{ \Gamma (r_1/2) \Gamma (r_2/2) } \int_0^\infty \left[ \int_0^
{(r_1/r_2)wv} \frac{ u^{r_1/2-1}e^{-u/2}}{2^{(r_1+r_2)/2}} du \right] v^{r_1/2-1}e^{-v/2} dv
\end{align*}\)

By differentiating the cdf , it can be shown that \(f(w) = F^\prime(w)\) is given by

\(\begin{align*}
f(w) = \frac{ \left( r_1/r_2 \right)^{r_1/2} \Gamma \left[ \left(r_1+r_2\right)/2 \right]w^{r_1/2-1} }
{\Gamma(r_1/2)\Gamma(r_2/2) \left[1+(r_1w/r_2)\right]^{(r_1+r_2)/2}}, \hspace{1cm} w>0
\end{align*}\)

A random variable with the pdf \(f(w)\) is said to have an F distribution with \(r_1\) and \(r_2\) degrees of freedom. We write this as \(F(r_1, r_2)\). Table VII in Appendix B of the textbook can be used to find probabilities for a random variable with the \(F(r_1, r_2)\) distribution.

It contains the F-values for various cumulative probabilities \((0.95, 0.975, 0.99)\) (or the equivalent upper − \(\alpha\)th probabilities \((0.05, 0.025, 0.01)\)) of various \(F (r1, r2)\) distributions.

When using this table, it is helpful to note that if a random variable (say, \(W\)) has the \(F(r_1, r_2)\) distribution, then its inverse \(\dfrac{1}{W}\) has the \(F(r_2, r_1)\) distribution.

The shape of the F distribution is determined by the degrees of freedom \(r_1\) and \(r_2\). The histogram below shows how an F random variable is generated using 1000 observations each from two chi-square random variables (\(U\) and \(V\)) with degrees of freedom 4 and 8 respectively and forming the ratio \(\dfrac{U/4}{V/8}\).

The lower plot (below histogram) illustrates how the shape of an F distribution changes with the degrees of freedom \(r_1\) and \(r_2\).