F distribution

Let $X$ and $Y$ be random variables such that

1. $X$ and $Y$ are independent
2. $X\sim \chi^2(m)$ , the chi-squared distribution with $m$ degrees of freedom
3. $Y\sim \chi^2(n)$ , the chi-squared distribution with $n$ degrees of freedom

Define a new random variable $Z$ by $$Z=\frac{(X/m)}{(Y/n)}.$$ Then the distribution of $Z$ is called the central F distribution, or simply the F distribution with m and n degrees of freedom, denoted by $Z\sim \operatorname{F}(m,n)$ .

By transformation of the random variables $X$ and $Y$ , one can show that the probability density function of the F distribution of $Z$ has the form: $$f_Z(x)=\frac{m^{m/2}n^{n/2}}{\operatorname{B}(\frac{m}{2},\frac{n}{2})} \cdot\frac{x^{(m/2)-1}}{(mx+n)^{(m+n)/2}},$$ for $x>0$ , where $\operatorname{B}(\alpha,\beta)$ is the beta function. $f_Z(x)=0$ for $x\le 0$ .

For a fixed $m$ , say 10, below are some graphs for the probability density functions of the F distribution with $(m,n)$ degrees of freedom.

The next set of graphs shows the density functions with $(m,n)$ degrees of freedom when $n$ is fixed. In this example, $n=10$ .

If $X\sim \chi^2(m,\lambda)$ , the non-central chi-square distribution with m degrees of freedom and non-centrality parameter $\lambda$ , with $Y$ and $Z$ defined as above, then the distribution of $Z$ is called the non-central F distribution with m and n degrees of freedom and non-centrality parameter $\lambda$ .

Remarks

• the ``F'' in the F distribution is given in honor of statistician R. A. Fisher.
• If $X\sim \operatorname{F}(m,n)$ , then $1/X\sim \operatorname{F}(n,m)$ .
• If $X\sim \operatorname{t}(n)$ , the t distribution with $n$ degrees of freedom, then $X^2\sim \operatorname{F}(1,n)$ .
• If $X\sim \operatorname{F}(m,n)$ , then $$\operatorname{E}[X] = \frac{n}{n-2}\mbox{ if }n>2,$$ and $$\operatorname{Var}[X] = \frac{2n^2(m+n-2)}{m(n-2)^2(n-4)}\mbox{ if }n>4.$$
• Suppose $X_1,\ldots,X_m$ are random samples from a normal distribution with mean $\mu_1$ and variance $\sigma_1^2$ . Furthermore, suppose $Y_1,\ldots,Y_n$ are random samples from another normal distribution with mean $\mu_2$ and variance $\sigma_2^2$ . Then the statistic defined by $$V=\frac{\hat{\sigma_1}^2}{\hat{\sigma_2}^2},$$ where $\hat{\sigma_1}^2$ and $\hat{\sigma_1}^2$ are sample variances of the $X_i's$ and the $Y_j's$ , respectively, has an F distribution with m and n degrees of freedom. $V$ can be used to test whether $\sigma_1^2=\sigma_2^2$ . $V$ is an example of an F test.