Given two independent random variables Y and Z.
\begin{align*}
Y &\sim \chi_m^2(Y=y) \\
Z &\sim \chi_n^2(Z=z) \\
\end{align*}
let
\begin{align*}
X = \frac{Y/m}{Z/n}
\end{align*}
then, X follows a f-distribution F_{m,n}(X=x) with (m, n) degrees of freedom.
\begin{align*}
X &\sim F_{m,n}(X=x) \\
&= \frac{m^{m/2} n^{n/2}}{\Beta(m/2, n/2)} \frac{x^{m/2 - 1}}{(mx + n)^{(m+n)/2}}
\end{align*}
let U = Z,
\begin{align*}
1 &= \int f_X(x) dx \\
&= \int \int f_{XU}(x, u) du dx \\
&= \int \int f_{YZ}(y, z) dy dz \\
&= \int \int f_Y(y) f_Z(z) dy dz \\
\end{align*}
y = \frac{m z x}{n} = \frac{m u x}{n} , z = u , x: 0 \rightarrow \infty , u: 0 \rightarrow \infty.
The Jaccobian is
\begin{align*}
|J|
&= \begin{vmatrix}
dy / dx & dy / du \\
dz / dx & dz / du \\
\end{vmatrix} \\
&= \begin{vmatrix}
\frac{m u}{n} & \frac{m x}{n} \\
0 & 1 \\
\end{vmatrix} \\
&= \frac{m u}{n}
\end{align*}
therefore,
\begin{align*}
&= \int_0^\infty
\int_0^\infty
f_Y(\frac{m u x}{n})
f_Z(u)
\frac{m u}{n}
dx du \\
&= \int_0^\infty
\int_0^\infty
\frac{1}{2^{\frac{m}{2}} \Gamma(\frac{m}{2})}
(\frac{m u x}{n})^{\frac{m}{2}-1}
e^{-\frac{\frac{m u x}{n}}{2}}
\frac{1}{2^{\frac{n}{2}} \Gamma(\frac{n}{2})} u^{\frac{n}{2}-1} e^{-\frac{u}{2}}
\frac{m u}{n}
dx du \\
&= \int_0^\infty
\frac{1}{
2^{\frac{m+n}{2}}
\Gamma(\frac{m}{2})
\Gamma(\frac{n}{2})
}
(\frac{m}{n})^{\frac{m}{2}}
x^{\frac{m}{2}-1}
\int_0^\infty
u^{\frac{m+n}{2}-1}
e^{-\frac{m u x + n u}{2n}}
du dx \\
\end{align*}
let \frac{(mx + n)u}{2n} = t, u = \frac{2 n t}{mx + n}, t: 0 \rightarrow \infty, du = \frac{2n}{m x + n} dt.
\begin{align*}
&= \int_0^\infty
\frac{1}{
2^{\frac{m+n}{2}}
\Gamma(\frac{m}{2})
\Gamma(\frac{n}{2})
}
(\frac{m}{n})^{\frac{m}{2}}
x^{\frac{m}{2}-1}
\int_0^\infty
(\frac{2 n t}{mx + n})^{\frac{m+n}{2}-1}
e^{-t}
\frac{2n}{m x + n} dt
dx \\
&= \int_0^\infty
\frac{1}{
\Gamma(\frac{m}{2})
\Gamma(\frac{n}{2})
}
(\frac{m}{n})^{\frac{m}{2}}
x^{\frac{m}{2}-1}
(\frac{n}{mx + n})^{\frac{m+n}{2}}
\int_0^\infty
t^{\frac{m+n}{2}-1}
e^{-t}
dt dx \\
&= \int_0^\infty
\frac{1}{
\Gamma(\frac{m}{2})
\Gamma(\frac{n}{2})
}
(\frac{m}{n})^{\frac{m}{2}}
x^{\frac{m}{2}-1}
(\frac{n}{mx + n})^{\frac{m+n}{2}}
\Gamma(\frac{m}{2} + \frac{n}{2}) dx \\
&= \int_0^\infty
\frac{1}{\Beta(\frac{m}{2}, \frac{n}{2})}
(\frac{m}{n})^{\frac{m}{2}}
x^{\frac{m}{2}-1}
(\frac{n}{mx + n})^{\frac{m+n}{2}}
dx \\
&= \int_0^\infty
\frac{1}{\Beta(\frac{m}{2}, \frac{n}{2})}
m^{\frac{m}{2}} n^{\frac{n}{2}}
x^{\frac{m}{2}-1}
\frac{1}{(mx + n)^{\frac{m+n}{2}}}
dx \\
&= \int_0^\infty
F_{m,n}(x)
dx \\
\end{align*}
Discussion