I was asked to prove that $Z^2 \sim \chi _1 ^2$ where $Z = \min (X,Y)$. Below is my working out: $$ F_{\mathcal{Z}^2}(z) := P(\mathcal{Z}^2 < z)$$ $$ = P(\mathcal{Z}^2 < z, X<Y) + P(\mathcal{Z}^2 < z, Y<X) $$ $$= P(\mathcal{Z}^2 < z | X<Y)\cdot P(X<Y) + P(\mathcal{Z}^2 < z | Y<X)\cdot P(Y<X)$$ $$= P(X^2 < z)\cdot \frac 12 + P(Y^2 < z)\cdot \frac 12$$ $$ = P(X^2 < z)\cdot \frac 12 + P(X^2 < z)\cdot \frac 12$$ $$ = P(X^2 < z)$$ $$= F_{X^2}(z)$$
My concern with my working out is with the second step. I am not sure that it is correct to split up the probability into the 2 cases where $X<Y$ and $Y<X$.