93. Prove that

(a) max(X1,X2) = X1 + X2 − min(X1,X2) and, in general,

(b) max(X1,...,Xn) =n 1

Xi −

i

+

i

+ (−1)

n−1 min(Xi,Xj ,...,Xn)

Show by defining appropriate random variables Xi, i = 1,...,n, and by taking expectations in part

(b) how to obtain the well-known formula P

n 1

Ai



=

i P(Ai) −

i

n−1 P(A1 ···An)

(c) Consider n independent Poisson processes—the ith having rate λi. Derive an expression for the expected time until an event has occurred in all n processes.