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Exercise 4 ((3 + 3) + 4 pts.). Conditional Distributions 59' Moment Generating Functions. Let X1,X2,. .. be i.i.d. random variables and N an independent nonnegative integer valued random variable. Let SN = X1 + . . . + XN. Assume that the m.g.f. of the Xi, denoted MX(t), and the m.g.f. of N, denoted M N(t) are nite in some interval (6, 6) around the origin. 1. (a) Express the m.g.f. MSN (t) of SN in terms of MX (t) and MN(t) Hint: Decompose the computation of the expectation by conditioning w.r.t. N. (b) Assume N N P0iss0n()\\) and X N Ber(p), deduce What is the distribution of SN from the previous question. 2. Compute the second moment ]E(SIQV) in terms of the moments of X and N and deduce the variance of S N in terms of the means and variances of X and N. Alternatively, if you had not found the solution of the rst question, use the following formula (coming from the practice problems for the exam): Var(X) = ]E[Var(X | Z)] +Var(]E[X | Z]) to give an expression of the variance of SN in terms of the means and variances of X and N