The question is suppose $X$ is a Poisson variable with a random mean, such that $X\sim \text{Poisson}(Y)$ and $Y \sim \exp(\lambda)$. Find the mean and variance of $X$.
By the law of total expectation, $E[X] = E[E[X|Y]]$ and I know that $E[X|Y]=E[X|\sigma(Y)]$. If I apply the definition of conditional expectation, I did not reach to any conclusion.
Is there any other way to find the expected value of $X$?