Problems when solving $E(X\mid Y)$

Question

If we know $X\sim \operatorname{Pois}(\lambda)$, $Y\sim\operatorname{Pois}(\lambda_p)$:

When solving $E(X\mid Y)$, based on the law of iterated expectations, $E(X) = E(E(X\mid Y)) =\lambda$. And we know that $E(\lambda) = \lambda$, so can we just say that $E(X\mid Y)=\lambda$ ? But this answer is wrong I suppose.

Answer 1

It is not possible in general to compute $E(X\mid Y)$ without any information on the dependence between $X$ and $Y$.

• If $X$ is independent of $Y$, then indeed, $E(X\mid Y)=\lambda$.
• If $\lambda>\lambda_p$ and $X=Y+Z$, where $Z$ is independent of $Y$ and has Poisson distribution with parameter $\lambda-\lambda_p$, then $E(X\mid Y)=Y+\lambda-\lambda_p$.