An insect lays a large number of eggs $N \sim \operatorname{Poi}(\Lambda), \Lambda>0$. Each of these eggs survives, independently of the other eggs, with probability $p \in (0,1)$. Now suppose we have a whole colony of egg-laying insects. Note that for each insect, the number of eggs laid can follow another Poisson distribution. Assume that $\Lambda \sim \operatorname{Exp}(\theta)$, where $\theta>0$ is constant. Calculate the mean number of survivors of a randomly selected insect.
Attempt: To calculate the expected number of survivors of a randomly selected insect, we need to calculate the conditional expectation of $Np$ given $\Lambda$ and then apply the law of total expectation.
The conditional probability of $Np$ given $\Lambda$ is simply a Poisson distribution with parameter $\Lambda p$. This is because the number of survivors for each egg is independent and follows a Poisson distribution with parameter $p$.
Therefore, the conditional expectation of $Np$ given $\Lambda$ is equal to $\Lambda p$. Using the law of total expectation, we can calculate the expected number of survivors of a randomly selected insect:
$$ \mathbb{E}(Np) = \mathbb{E}(\mathbb{E}(Np|\Lambda)) = \mathbb{E}(\Lambda p) = p \mathbb{E}(\Lambda) = p \frac{1}{\theta}. $$
Therefore, the expected number of survivors of a randomly selected insect is equal to $p/\theta$.