F-distribution

$$X=\frac{\frac{1}{d_1}\sum_{i=1}^{d1} Z_i^2}{\frac{1}{d_2}\sum_{i=1}^{d2} Z_i^2}$$

$$f(x) = \frac{\sqrt{\frac{(d_1x)^{d_1}d_2^{d_2}}{(d_1x+d_2)^{d_1+d_2}}}}{x\text{B}\big(\frac{d_1}{2}, \frac{d_2}{2}\big)} \\[24pt] \text{where $d_1$ and $d_2$ are the degrees of freedom of the two} \\ \\ \text{chi-square random variables, and $\text{B}$ is the beta function.}$$