Question: Problem 4 (24 points) a. Suppose N and N2 are independent Poisson-distributed random variables with parameters , and 12, respectively. Let n be some positive

Problem 4 (24 points) a. Suppose N and N2 are independent Poisson-distributed random variables with parameters , and 12, respectively. Let n be some positive integer. Derive the conditional pmf of N1 given N1 + N2 = n. Can you identify this distribution? b. Let r1, 12 E (0, oo) and p e (0, 1). Let Ni ~ NegBinomial(r1, p) and N2 ~ NegBinomial(r2, p). Take n to be a positive integer and k E {1, 2, ..., n} and show P(N1 = k(N1 + N2 = n) = n T( k ta)I(n - k+ B) [(a+ 3) k I(n + at B) I(a) I(B) for some appropriate values a, B E (0, co)

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