Definition
Let U∼χ2(r1),V∼χ2(r2) be independent random variables following Chi-squared distributions, then
V/r2U/r1∼F(r1,r2)
where r1,r2∈N are the degrees of freedoms
Properties
PDF
{\Gamma(\frac{r_{1} + r_{2}}{2}) (\frac{r_{1}}{r_{2}})^{\frac{r_{1}}{2}} x^{\frac{r_{1}}{2}-1}}
{\Gamma(\frac{r_{1}}{2}) \Gamma(\frac{r_{2}}{2}) (\frac{r_{1}}{r_{2}}x+1)^{(r_{1} + r_{2})/2}}$$
## Mean
$$E(X) = \frac{r_{2}}{r_{2} - 2},\quad r_{2}>2$$
## Variance
$$\operatorname{Var}(X) = \frac{2(r_{1}+r_{2}-2)}{r_{1} (r_{2}-2)^{2}(r_{2}-4)},\quad r_{2}>4$$