If X and Y are independent Poisson r.v.'s with respective means $\lambda_X$ and $\lambda_Y$, find $E[X\mid X+Y\neq n]$

Question

If X and Y are independent Poisson r.v.'s with respective means $\lambda_X$ and $\lambda_Y$, find $E[X\mid X+Y\neq n]$

I have seen a similar problem where the condition is that $X+Y = n$. Then resulting pmf turns out to be binomial with parameters $n$ and $\frac{\lambda_X}{\lambda_X+\lambda_Y}$ so it's easy to find the expected value which is the product of the two parameters.

Since the condition this time is that $X+Y\neq n$, I'm thinking of approaching this somehow using the complement of the previous pmf, but I don't know if that is mathematically correct to do it that way. Any suggestions?

Answer 1

$$ \mathsf{E}[X]=\mathsf{E}[X\mid X+Y=n]\mathsf{P}(X+Y=n)+\mathsf{E}[X\mid X+Y\ne n]\mathsf{P}(X+Y\ne n) $$

Thus, $$ \mathsf{E}[X\mid X+Y\ne n]=\frac{\lambda_X-\mathsf{E}[X\mid X+Y=n]\mathsf{P}(X+Y=n)}{1-\mathsf{P}(X+Y=n)}. $$