Suppose that $X_{1}, \ldots, X_{n}$ are a random sample from a distribution with density $f(x)$. Prove that the joint density (pdf) of $Y=\min -1 \leq i \leq n) X_{i}$ and $2=\max _{1 \leq i \leq n} X_{i}$ is $$ f(y, z)=\left\{\begin{array}{11} n(n-1) {F(z)-F(y) }^{n-2} f(y) f(z), & ycz 0 & \text { otherwise } \end{array} ight. $$ Hint: Compute $P(y \leq Y, Z \leq z) $ S.P.PB.212