8 The sum of two independent Poissons is Poisson Let X Y be

8. The sum of two independent Poissons is Poisson Let X, Y be two independent Poisson random variables with parameters and µ, respectively.

(a) Show that X +Y is also Poisson with parameter +µ, in other words, show that P(X + Y = k) = e (+µ) ( + µ) k k! , k = 0, 1 . . . Hint: Start with the observation that {X + Y = k} = [ k m=0 {X = m, Y = k m} and apply the addition rule. Recall also the Binomial expansion.

(b) Show that the conditional pmf of X given X +Y = n is Binomial with parameters n and p = /( + µ), i.e., P(X = k|X + Y = n) = n k + µ k µ + µ nk , k = 0, . . . , n.

(c) What is E[X|X + Y = n] and Var[X|X + Y = n]?

Solution