# 23.3 - F Distribution

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.

## Illustration

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 $$r_2$$.

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