59. Prove that (a) maxC^, X2) = Xx + X2 - min^, X2) and, in general, ...

59. Prove that

(a) maxCÄ^, X2) = Xx + X2 - min^, X2) and, in general,

ç

(b) maxi*! , . . . , * „ ) = Ó Ë, - Ó Ó m i n( * « . Xj)

1 i

+ Ó Ó Ó mm(Xi,Xj,Xk) + ...

i

+ (-l)"-1min(X1,X2,...,Xn)

Show by defining appropriate random variables Xi9 i = 1 , n , and by taking expectations in

(b) how to obtain the well-known formula p((jA) = Ó PiAi) - Ó Ó P(4iAj)
\ 1 / i i

(c) Consider ç independent Poisson processes—the /th having rate ë,.
Derive an expression for the expected time until an event has occurred in all ç processes.