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. What is the conditional expectation E[X∣X+Y=n]E[X \mid X+Y = n]E[X∣X+Y=n]?
nλ1λ1+λ2n \frac{\lambda_1}{\lambda_1 + \lambda_2}nλ1+λ2λ1
nλ2λ1+λ2n \frac{\lambda_2}{\lambda_1 + \lambda_2}nλ1+λ2λ2
λ1λ1+λ2\frac{\lambda_1}{\lambda_1 + \lambda_2}λ1+λ2λ1
λ1\lambda_1λ1