Suppose I have a fruit stand where I sell oranges. Every day I have an average of $\lambda$ people (Poisson distributed) each buying a normally distributed weight quantity of oranges with mean $\mu$ and standard deviation $\sigma$.
What's the mean and variance of the number of pounds of oranges I sell per day?
OK, the mean is trivial: $\lambda \cdot \mu$. But how do I calculate the variance?
Strangely enough I've come up with an answer through trial and error that seems to fit really well with my Monte Carlo simulations, but I can't convince myself that it's actually the correct answer in theory. If someone can point me to the "right" answer, I'll post the one I got empirically for comparison.