(a) [10 points] Prove following Theorem 6.4 in the GIP Textbook Chapter 3, page 85. Theorem 6.4. Let X1, X2, ... be i.i.

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A 10 Points Prove Following Theorem 6 4 In The Gip Textbook Chapter 3 Page 85 Theorem 6 4 Let X1 X2 Be I I 1 (85.15 KiB) Viewed 95 times

(a) [10 points] Prove following Theorem 6.4 in the GIP Textbook Chapter 3, page 85. Theorem 6.4. Let X1, X2, ... be i.i.d. random variables, and let N be a nonnegative, integer-valued random variable independent of X1, X2, .... Set So = 0 and Sn = X1 + X2 + ... + Xn, for n > 1. Then 48x (t) = 9N (4x(t)). (b) [10 points] Use the above Theorem 6.4 to prove the results in the following Theorem 6.2 in the GIP Textbook Chapter 3, page 81 for general i.i.d. Xị, that are not necessarily integer-valued, but can be either discrete or continuous distribution. Theorem 6.2. Suppose that the conditions of Theorem 6.1 are satisfied. (a) I, moreover, EN < and E|X|<0, then ESN = ENEX (b) If, in addition, Var N < 0 and Var X < 0, then Var Sn = EN. Var X + (EX)2. Var N.