# 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) ?

Thank you in advance !

• 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
• Welcome to CV. Since you’re new here, you may want to take our tour, which has information for new users. Please add the [self-study] tag and read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. If this is self-study rather than homework, let us know, and... it's still a good idea to show us what you've tried or explain specifically what you don't understand. – jbowman Jan 22 at 5:21
• This was an exam question I couldn't answer properly. I know that the number of customers arriving at that cashpoint after t hours is distributed Poisson(tlambda). The expectation of t(x), therefore must be equal to tlambda*E(Yi). However, I couldn't figure out what the variance of t(x) would be. I will be more clear next time I make a post and will add self-study tag. Thank you – Urve Görkem Taşcı Jan 22 at 10:41