Expected daily revenue from random shoppers

Question

Customers arrive at a shop as a Poisson process at rate $20$ per hour. Independently, each customer spends an amount averaging $\$ 15 $. The shop is open for$ 8$ hours.

What is the expected total revenue for the day?

Accepted Answer

When a random number of arrivals each carry a random amount, the running total is a compound Poisson sum. Its expected value factors into a beautifully simple product: the expected number of arrivals times the average amount each one contributes. First, the expected number of customers in the day is E[N] = \lambda t = 20 	imes 8 = 160. Each spends \15 on average, so the expected total revenue is E[	ext{revenue}] = E[N] 	imes E[	ext{spend}] = 160 	imes 15 = \2400. The reasoning is just: on an average day 160 people walk in, and each is worth about \15, so the store expects about \2400. (T…

Expected daily revenue from random shoppers

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