Let 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) be independent. Find the conditional expectation E[X(X−1)∣X+Y=n]E[X(X-1) \mid X+Y = n]E[X(X−1)∣X+Y=n].
n(n−1)(λ1λ1+λ2)2n(n-1) \left( \frac{\lambda_1}{\lambda_1 + \lambda_2} \right)^2n(n−1)(λ1+λ2λ1)2
n(n−1)λ1λ1+λ2n(n-1) \frac{\lambda_1}{\lambda_1 + \lambda_2}n(n−1)λ1+λ2λ1
n2(λ1λ1+λ2)2n^2 \left( \frac{\lambda_1}{\lambda_1 + \lambda_2} \right)^2n2(λ1+λ2λ1)2
λ12\lambda_1^2λ12