Transforming of Chi Square Distribution and F Distribution

Question

Let X and Y be two independent Chi-Square-distributed radom variables each with n d.f. Then $ Z = \frac{\sqrt n}{2} \frac{X - Y}{\sqrt {X.Y}}$ has the t-Distribution. (I proved this already).
But I can't prove how Z can be written as $ Z = \frac{\sqrt n}{2} (\sqrt{F_{n,n}}-\frac{1}{\sqrt{F_{n,n}}})$ with $F_{n,n}$ is F distribution with n and n d.f. Someone can help me please. Thanks in advance!

Answer 1

If $X$ has chi-square($m$) distribution and $Y$ has chi-square($n$) distribution, and the two are independent, then it is either a theorem or a definition that $(X/m)\big/(Y/n)$ has $F(m,n)$ distribution. So defining $W:=X/Y$, we have $$Z=\frac{\sqrt n}2\left(\sqrt W-\frac1{\sqrt W}\right).$$