One other question I came across that I didn't quite understand.
The number of forks that enter the sink follows a Poisson process with rate $λ= 200$ per month. Each fork which enters the sink has probability $p$ of smashing into pieces, otherwise it remains intact. Assume that each fork smashes upon impact or not after entering the sink independently of everything else.
Suppose that the mass of any fork that smashes in the sink is independently drawn from an Exponential distribution with mean $10$ g. Let $Y$ be the total mass of forks that smash upon impact in the sink in a year. Calculate the mean and variance of $Y$. (Assume that one year equals exactly 12 months)
This is almost an extension of my last question, which I found in my text, that I'm trying to do for practice.
My thinking is that I should be using the compound random variable formula, and that I should be letting $X_{i}$ be the mass of any given fork which smashes in the sink. But I'm not really sure how I should explicitly express my problem, nor how to find the distribution for $Y$ in order to calculate the mean/variance.
Let $X$ represent the number of forks that enter the sink in a year.
$$X \sim \text{Poi}(2400) $$
$Z$ represent the number of smashed forks in a year.
$$X \sim \text{Poi}( 2400 p) $$
$Y$ represent the mass of smashed forks in a year.
$$Y = E_1 + \ldots + E_Z $$
where $E_i$ is are independent exponential random variable of mean $10$ and variance $100$.
Therefore $$E(Y) = E(Z E_1) = E(Z) E(E_1) = 24000 p $$
and $$Var(Y) = E(Z^2 E_1^2) - E(Z E_1)^2 = ((2400p)^2 + 2400p)110 - (24000 p)^2 $$