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(a) The sales department of certain company receives purchase orders by either of two methods: phone or fax. Let X and Y denote the number of phone orders and fax orders, respectively, received in a fixed time period. Assume that both X and Y have Poisson distributions with parameters alpha and beta, respectively. Further, assume that X and Y are independent. Prove that X + Y has a Poisson distribution and give its parameter. (b) Find the conditional distribution for X given that X + Y = n, and give its mean and variance.

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