UberEats is considering a subscription service in which its customers pay a flat rate for as many food deliveries as they want during a given month. Suppose UberEats's internal costs for Delivery i (driver salary/tip, insurance, gas, etc.) is Xi , a uniform random variable on [4, 20] dollars. Let Yi be the (continuous) number of days between a customer's Delivery i 1 and Delivery i, modeled as an exponential random variable with mean 6 days (here, Y1 is the time until the first delivery of the month). For the purposes of this problem, assume the deliveries are instantaneous, and that all the {Xi} and {Yi} are mutually independent. Let N be the number of orders a customer makes over the course of 30 days, and let Z be the internal monthly cost to UberEats for delivering these N orders. Note that N is a discrete random variable (we can't make half a delivery!).

(a) (7 points) Determine the conditional expected value E(Z | N).

(b) (7 points) Determine the expected value of Z. Remember the connection between the exponential and Poisson random variables!

(c) (6 points) How much should UberEats charge for its monthly service to make an average of 15 dollars net profit per month from each customer?