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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 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 $Z\sim\operatorname{F}(m,n)$.
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 $x\leq 0$.
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 $X\sim\operatorname{F}(m,n)$, 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
$\operatorname{E}[X]=\frac{n}{n2}\mbox{ if }n>2,$ and
$\operatorname{Var}[X]=\frac{2n^{2}(m+n2)}{m(n2)^{2}(n4)}\mbox{ if }n>4.$ 
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 $X_{i}^{{\prime}}s$ 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.
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Comments
nice graphs
and done with pstricks, even better ;)
f
G > H G
p \ /_  ~ f(G)
\ / f ker f
G/ker f
Re: nice graphs
Thanks! And thanks for all the help drini! Chi