the distribution of $\frac{(X_1 +X_2)^2}{(X_1 -X_2)^2}$

Question

If $X_1$ and $X_2$ are random sample of size $2$ from a $N(0, 1)$ population, then the distribution of $\frac{(X_1 +X_2)^2}{(X_1 -X_2)^2}$

My work: I find the expectation of $E(X_1 +X_2)^2 = E(X_1 -X_2)^2 =2$, but after that how do I proceed?

Answer 1

Observe that since $X_1$ and $X_2$ are indepedent $$ X_1+X_2\sim N(0,2);\quad X_1-X_2\sim N(0,2) $$ where the second parameter is variance. We may write $X_1+X_2\stackrel{d}{=}\sqrt{2}X_1$ and $X_1+X_2\stackrel{d}{=}\sqrt{2}X_2$. Hence $$ \frac{(X_1 +X_2)^2}{(X_1 -X_2)^2}\stackrel{d}{=}\frac{X_1^2}{X_2^2}\sim F(1,1) $$ because $X_1^2\perp X_2^2$ and $X_1^2\sim \chi^2_{(1)}$.

Note $\stackrel{d}{=}$ means equal in distribution.