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Ratio of 2 independent chi square random variables divided by their degrees of freedom is F distribution

When V and U are two \(\chi^2\) independent random variables: \(f_V(v)=\frac{(\frac{1}{2})^{\frac{m}{2}}}{\Gamma(\frac{m}{2})}v^{(\frac{m}{2})-1}e^{-\frac{1}{2}v}\)
\(f_U(u)=\frac{(\frac{1}{2})^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}u^{(\frac{n}{2})-1}e^{-\frac{1}{2}u}\)
with \(m\) and \(n\) degrees of freedom, then, the pdf for \(W=V/U\) is:
\[\begin{align} f_{V/U}(\omega)&=\int_{0}^{+\infty}|u|f_U(u)f_V(u\omega)du\\ &=\int_{0}^{+\infty}u\frac{(\frac{1}{2})^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}u^{\frac{n}{2}-1}e^{-\frac{1}{2}u} \frac{(\frac{1}{2})^{\frac{m}{2}}}{\Gamma(\frac{m}{2})}(u\omega)^{\frac{m}{2}-1}e^{-\frac{1}{2}u\omega}du\\ &=\frac{(\frac{1}{2})^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}\frac{(\frac{1}{2})^{\frac{m}{2}}}{\Gamma(\frac{m}{2})} \omega^{\frac{m}{2}-1} \int_{0}^{+\infty}u^{\frac{n}{2}}u^{\frac{m}{2}-1} e^{-\frac{1}{2}u(1+\omega)}du\\ &=\frac{(\frac{1}{2})^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}\frac{(\frac{1}{2})^{\frac{m}{2}}}{\Gamma(\frac{m}{2})} \omega^{\frac{m}{2}-1} \int_{0}^{+\infty}u^{\frac{n+m}{2}-1} e^{-\frac{1}{2}u(1+\omega)}du\\ &=\frac{(\frac{1}{2})^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}\frac{(\frac{1}{2})^{\frac{m}{2}}}{\Gamma(\frac{m}{2})} \omega^{\frac{m}{2}-1} (\frac{\Gamma(\frac{n+m}{2})}{(\frac{1}{2}(1+\omega))^{\frac{n+m}{2}}})\\ &=\frac{\Gamma(\frac{n+m}{2})}{\Gamma(\frac{n}{2})\Gamma(\frac{m}{2})}\frac{\omega^{\frac{m}{2}-1}}{(1+\omega)^{\frac{n+m}{2}}} \end{align}\]

Then, the pdf for \(W=\frac{V/m}{U/n}\) is: \[\begin{align} f_{\frac{V/m}{U/n}}&=f_{\frac{n}{m}V/U}\\ &=\frac{m}{n}f_{V/U}(\frac{m}{n}\omega)\\ &=\frac{m}{n}\frac{\Gamma(\frac{n+m}{2})}{\Gamma(\frac{n}{2})\Gamma(\frac{m}{2})}\frac{(\frac{m}{n}\omega)^{\frac{m}{2}-1}}{(1+\frac{m}{n}\omega)^{\frac{n+m}{2}}}\\ &=\frac{\Gamma(\frac{n+m}{2})}{\Gamma(\frac{n}{2})\Gamma(\frac{m}{2})}\frac{m}{n}\frac{(\frac{m}{n}\omega)^{\frac{m}{2}-1}}{(n+m\omega)^{\frac{n+m}{2}}}n^{\frac{n+m}{2}}\\ &=\frac{\Gamma(\frac{n+m}{2})}{\Gamma(\frac{n}{2})\Gamma(\frac{m}{2})}\frac{m^{\frac{m}{2}}n^{\frac{n}{2}}\omega^{\frac{m}{2}-1}}{(n+m\omega)^{\frac{n+m}{2}}} \end{align}\], which is a \(F\) distribution with \(m\) and \(n\) degrees of freedom.