Let X Poisson(), Y Poisson(), independent It was shown in Theorem 4 3 2 that the distribution of X Y is Poisson( ) Show that the distribution of X X Y is binomial with success probability ( ) What is the distribution of Y X Y

Question: Let X ~ Poisson(), Y ~ Poisson(), independent. It was shown in Theorem 4.3.2 that the distribution of X + Y is Poisson( + ).

Let X ~ Poisson(θ), Y ~ Poisson(λ), independent. It was shown in Theorem 4.3.2 that the of X + Y is Poisson(θ + λ). Show that the of X|X + Y is binomial with success probability θ/(θ + λ). What is the of Y|X + Y?

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