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Distributionshard
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Let XXX and YYY be independent random variables such that X∼Poisson(λ1)X \sim \text{Poisson}(\lambda_1)X∼Poisson(λ1​) and Y∼Poisson(λ2)Y \sim \text{Poisson}(\lambda_2)Y∼Poisson(λ2​). What is the conditional distribution P(X=k∣X+Y=n)P(X = k | X+Y = n)P(X=k∣X+Y=n)?