(i) Let X1, X2, Be r v s and let N be a r v taking the values 1, 2, , all defined on the probability on the probability space ((, A, P) Define the function X as X (() X1 (() X N (() ((), And show that X is a r v (ii) Now suppose that the Xi s are independent and identically distributed with X1 ( (, that N is independent of the Xi s, and that N ( Then show that (X N) N, and therefore X (N) (iii) If in addition to the assumptions made in part (ii), it also holds that Var (X1) (2 ( and Var(N) (, then show that Var(X N) (2 N (iv) Use parts (ii) and (iii) here and part (ii) of Exercise 16 in order to conclude that Var(X) (2 (N) (2 Var(N)

Question: (i) Let X1, X2,.... Be r.v.s and let N be a r.v. taking the values 1, 2,..., all defined on the probability on the probability

(i) Let X1, X2,.... Be r.v.s and let N be a r.v. taking the values 1, 2,..., all defined on the probability on the probability space ((, A, P). Define the function X as
X (() = X1 (() + ... + X N (() ((),
And show that X is a r.v.
(ii) Now suppose that the Xi s are independent and identically distributed with εX1 = μ ( (, that N is independent of the Xi s, and that εN < (. Then show that ε(X | N) = μN, and therefore εX = μ(εN).
(iii) If in addition to the assumptions made in part (ii), it also holds that Var (X1) = (2 < ( and Var(N) < (, then show that Var(X | N) = (2 N.
(iv) Use parts (ii) and (iii) here and part (ii) of Exercise 16 in order to conclude that Var(X) = (2 (εN) + (2 Var(N)?

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