I have a question that reads: "An insurance company has initial capital of £$20,000$ and receives claims from clients with a homogeneous Poisson process $X(t),t\geq 0$. The intensity of claims is $6$ claims a month. Assume that the amount $Z_k,k=1,2,...$ of successive claims are independent random variables with common cumulative distribution function $F(x) = P${$Z_k \leq x$} = $1-e^{-(1/200) x},x\geq 0$. Suppose the random variables $Z_k$ are independent of $X(t)$. Also assume all 12 months in the year have 30 days. Find the mean and standard deviation of total capital of the insurance company during one year if the premium rate is £$3000$ a month."
I'm a bit puzzled here because usually when I do questions like this the mean is provided and we are asked to find the ruin probability. I know $\mu = EZ_k$ but how do we find that here?
I tried $\int^{\infty}_0 x(1-e^{-(1/200) x})$ but this diverges and I don't get a solution