Chapter 6, Problem 6.33TE

Let $X_1,\mathrm{...},X_n$ be a set of independent and identically distributed continuous random variables having distribution function F and let $X_{\left(i\right)},i=1,\mathrm{...},n$ denote their ordered values. If X, independent of the $X_i,i=1,\mathrm{...},n$ also has distribution F, determine

a. $P\left\{X>X_{\left(n\right)}\right\}$;

b. $P\left\{X>X_{\left(1\right)}\right\}$;

c. $P\left\{X_{\left(i\right)}<X<X_{\left(j\right)}\right\},1\le i<i\le n$.