# F distribution

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

1. 1.
2. 2.

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

3. 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 $x\leq 0$.

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 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

Title F distribution FDistribution 2013-03-22 14:26:56 2013-03-22 14:26:56 CWoo (3771) CWoo (3771) 15 CWoo (3771) Definition msc 62A01 Fisher F distribution F-distribution central F-distribution central F distribution non-central F distribution