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 (http://planetmath.org/ChiSquaredRandomVariable) 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 $Z\sim \mathrm{F}(m,n)$.

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:

$$f_Z(x)=\frac{m^{m/2}{n}^{n/2}}{\mathrm{B}(\frac{m}{2},\frac{n}{2})}\cdot \frac{x^{(m/2)-1}}{{(mx+n)}^{(m+n)/2}},$$

for $x>0$, where $\mathrm{B}(\alpha ,\beta )$ 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-centralF 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 $X\sim \mathrm{F}(m,n)$, then $1/X\sim \mathrm{F}(n,m)$.

• If $X\sim \mathrm{t}(n)$, the t distribution with $n$ degrees of freedom, then $X^2\sim \mathrm{F}(1,n)$.

• If $X\sim \mathrm{F}(m,n)$, then

  $$\mathrm{E}[X]=\frac{n}{n-2}\text{ if }n>2,$$

  and

  $$\mathrm{Var}[X]=\frac{2n^2(m+n-2)}{m{(n-2)}^2(n-4)}\text{ if }n>4.$$

• Suppose $X_1,\mathrm{\dots },X_m$ are random samples from a normal distribution with mean ${\mu }_1$ and variance ${\sigma }_1^2$. Furthermore, suppose $Y_1,\mathrm{\dots },Y_n$ are random samples from another normal distribution with mean ${\mu }_2$ and variance ${\sigma }_2^2$. Then the statistic defined by

  $$V=\frac{{\widehat{{\sigma }_1}}^2}{{\widehat{{\sigma }_2}}^2},$$

  where ${\widehat{{\sigma }_1}}^2$ and ${\widehat{{\sigma }_1}}^2$ are sample variances of the $X_i^{\prime }s$ and the $Y_j^{\prime }s$, 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.