PlanetMath: F distribution

(more info)

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F distribution

(Definition)

Let and be random variables such that

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

Define a new random variable

$\displaystyle Z=\frac{(X/m)}{(Y/n)}.$

Then the distribution of

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 and , one can show that the probability density function of the F distribution of has the form:

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

for

, where $\operatorname{B}(\alpha,\beta)$ is the beta function.

for $x\le 0$ .

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

$\includegraphics[scale=0.9]{fdist1}$

The next set of graphs shows the density functions with degrees of freedom when is fixed. In this example, .

$\includegraphics[scale=0.9]{fdist2}$

If $X\sim \chi^2(m,\lambda)$ , the non-central chi-square distribution with m degrees of freedom and non-centrality parameter $\lambda$ , with and defined as above, then the distribution of 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 degrees of freedom, then $X^2\sim \operatorname{F}(1,n)$ .
If $X\sim \operatorname{F}(m,n)$ , then
$\displaystyle \operatorname{E}[X] = \frac{n}{n-2}$ if $\displaystyle n>2,$
and
$\displaystyle \operatorname{Var}[X] = \frac{2n^2(m+n-2)}{m(n-2)^2(n-4)}$ if $\displaystyle n>4.$
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
$\displaystyle V=\frac{\hat{\sigma_1}^2}{\hat{\sigma_2}^2},$
where $\hat{\sigma_1}^2$ and $\hat{\sigma_1}^2$ are sample variances of the and the , respectively, has an F distribution with m and n degrees of freedom. can be used to test whether $\sigma_1^2=\sigma_2^2$ . is an example of an F test.