F distribution
Let $X$ and $Y$ be random variables^{} such that

1.
$X$ and $Y$ are independent^{}

2.
$X\sim {\chi}^{2}(m)$, the chisquared distribution (http://planetmath.org/ChiSquaredRandomVariable) with $m$ degrees of freedom

3.
$Y\sim {\chi}^{2}(n)$, the chisquared 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 noncentral chisquare distribution with m degrees of freedom and noncentrality parameter $\lambda $, with $Y$ and $Z$ defined as above, then the distribution of $Z$ is called the noncentral F distribution with m and n degrees of freedom and noncentrality 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
$$\mathrm{E}[X]=\frac{n}{n2}\text{if}n2,$$ and
$$\mathrm{Var}[X]=\frac{2{n}^{2}(m+n2)}{m{(n2)}^{2}(n4)}\text{if}n4.$$ 
•
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.
Title  F distribution 

Canonical name  FDistribution 
Date of creation  20130322 14:26:56 
Last modified on  20130322 14:26:56 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  15 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 62A01 
Synonym  Fisher F distribution 
Synonym  Fdistribution 
Synonym  central Fdistribution 
Synonym  central F distribution 
Defines  noncentral F distribution 