# Poisson Process Simple Question

The number of customers that arrive at a cashpoint in an hour is distributed poisson($$\lambda$$). Suppose that each arriving customer makes a draft. Let $$Y_i$$ denote the amount of money $$i^{th}$$ customer cashes. We know that the expectation of $$Y_i$$ is equal to 30 dollars and the variance of $$Y_i$$ is equal to 10 dollars.

Let t be the number of hours and x(t) be the total amount of money that is drawn from the cashpoint at the end of t hours. Then, what is the expected value and the variance of x(t) ?

• General topic is sometimes called 'Random Sum of Random Variables'. Notice that $X_t$ has two sources of variability. – BruceET Jan 21 at 17:39