4.5.1 $F$-Distribution

Theorem

Let $U$ and $V$ be two independent random variables having chi-squared distributions with $v_1$ and $v_2$ df, respectively.

Then the distribution of the random variable $F=\frac{U/v_1}{V/v_2}$ is given by the density function $h(f) = \left \{ \frac{\Gamma(v_1+v_2)/2^{\frac{v_1}2}}{\Gamma(\frac{v_1}2)\Gamma(\frac{v_2}2)}\frac{f^{\frac{v_1}2)-1}}{(1+v_1\frac{f}2)^{\frac{v_1+v_2}2}} \right .$

$$

h(f) = \left \{ \begin{align*} \frac{\Gamma(v_1+v_2)/2^{\frac{v_1}2}}{\Gamma(\frac{v_1}2)\Gamma(\frac{v_2}2)} &\frac{f^{\frac{v_1}2)-1}}{(1+v_1\frac{f}2)^{\frac{v_1+v_2}2}} &; &\ f \gt 0\\ &0 &; &\ f \le 0 \end{align*} \right .

$$

• $F$-Distribution Table

Corollary

If $S^2_1$ and $S^2_2$ are the variances of independent random samples of size $n_1$ and $n_2$ taken from normal populations with variances $\sigma^2_1$ and $\sigma^2_2$ respectively, then

$F =\frac{S^2_1/ \sigma^2_1}{S^2_2/\sigma^2_2} =\frac{\sigma^2_2S^2_1}{\sigma^2_1S^2_2/}$ has a $F$-distribution with $v_1=n-1$ and $v_2=n-1$ df.

4.5.2 How does the $F$-Distribution look like?

• Density function of $F$ depends on $v_1,v_2$ but also on the order that we state them

• The $\alpha$ value in statistical table of $f_\alpha$ is usually given by $a = 0.05$ or $a = 0.01$. (See the tables in 4.5.1)

Theorem

Writing $f_a(v_1,v_2)$ for $f_a$ with $v_1$ and $v_2$ degrees of freedom, we have

$$ f_{1-a}(v_1,v_2) = \frac{1}{f_a(v_2,v_1)} $$

e.g. $f_{0.95}(6,10)=\frac{1}{f_{0.05}(10,6)}$

4.5.3 What is the $F$-distribution Used For?

The $F$-distribution is also called the variance ratio distribution. It is generally used for the analysis of variance.