Problems when solving $E(X\mid Y)$
Solution 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 $E(X\mid Y)=Y+\lambda-\lambda_p$.