7. Let X and Y be independent random variables having Poisson distributions with parameters and , respectively. Prove that X + Y has a

7. Let X and Y be independent random variables having Poisson distributions with parameters

λ and μ, respectively. Prove that X + Y has a Poisson distribution and that var(X + Y ) = var(X) + var(Y ). Find the conditional probability P(X = k | X + Y = n) for 0 ≤ k ≤ n, and hence show that the conditional expectation of X given that X + Y = n, that is, E

????

X X + Y = n

=

∞X k=0 kP

????

X = k X + Y = n

, is nλ/(λ + μ). (Oxford 1983M)