Question: Let \({{X}_{1}},{{X}_{2}},....,{{X}_{n}}\) be independent random variables, and all are \(U\left( 0,1 \right)\). Define

\[X=\max \left\{ {{X}_{1}},...,{{X}_{n}} \right\}\] \[V=\min \left\{ {{X}_{1}},{{X}_{2}},...,{{X}_{n}} \right\}\]

(a) Compute \(E\left( \max \left\{ {{X}_{1}},{{X}_{2}} \right\} \right)\) and \(E\left( \min \left\{ {{X}_{1}},{{X}_{2}} \right\} \right)\)

(b) Compute \(E\left( X \right)\) and \(E\left( V \right)\) in general.

(c) Can you argue directly that \(1-E\left( \max \left\{ {{X}_{1}},...,{{X}_{n}} \right\} \right)=E\left( \min \left\{ {{X}_{1}},...,{{X}_{n}} \right\} \right)\)