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224Compound Poisson Distribution: MGF and MomentsLet N \sim Poisson ( ) and let X 1, X 2, \ldots be iid discrete random variables (independent of N) with PMF P(X i = j) = p j for j = 1, 2, \ldots and MGF M X(t) = E[e tX 1 ]. Define the **compound Poisson** sum S = \sum i=1 N X i (with S = 0 when N = 0). (a) Derive the MGF of S. Show that M S(t) = \exp\!\big( (M X(t) - 1)\big). (b) Use the MGF to derive E[S] and Var (S). Express your answers in terms of , E[X 1], and Var (X 1). (c) Alternatively, derive E[S] and Var (S) using the tower property (law of total expectation) and the Eve's law (law of total variance), conditioning on N. (d) **Application:** An insurance company receives claims at a Poisson rate of = 10 per day. Each claim size is \1000 with probability 0.6 or \5000 with probability 0.4, independently. Find E[S] and Var (S) for the total daily claims S, and compute the standard deviation.概率困难derivation未尝试免费225Minimum of Independent Geometric Random VariablesLet X 1, X 2, \ldots, X n be independent, each X i \sim Geometric (p i) with P(X i = k) = (1-p i) k-1 p i for k = 1, 2, \ldots (the "number of trials until first success" convention). Define M = \min(X 1, \ldots, X n). (a) Show that P(M > k) = \prod i=1 n (1-p i) k for k = 0, 1, 2, \ldots (b) Prove that M \sim Geometric \! (1 - \prod i=1 n (1-p i) ). State the PMF of M explicitly. (c) For the iid case p i = p for all i: express E[M] and Var (M) in terms of n and p, and verify that E[M] 1 as n . (d) **Application:** Five independent traders each attempt to fill an order on any given day with probability 0.3. What is the expected number of days until the first fill occurs? What is the probability that no fill occurs in the first 3 days? (e) Show that P(X j = M \mid M = m) depends on j (when p i are not all equal) and compute this probability for n = 2, p 1 = 0.3, p 2 = 0.5, m = 2.概率困难derivation未尝试免费