Let $X \sim N (0, 1)$ and $Y ∼ N (0, 1)$ be two independent random variables, and define $Z = \min(X, Y )$. Prove that $Z^2\sim\chi^2(1),$ i.e. Chi-Squared with degree of freedom $1.$
I found the density functions of $X$ and $Y,$ as they are normally distributed. How would one use the fact that $Z = \min(X,Y)$ to answer the question? Thanks!