A random number of customers, each an average spend

Question

A stall gets a random number of customers each day. The count averages $40$ , but on any given day it could be more or less. Each customer, independently of the others and of how many show up, spends an amount averaging $\$ 12$.

What daily revenue should you expect?

Accepted Answer

Let N be the number of customers and let the total revenue be the sum of their spends. Both the count and the individual spends are random. Condition on N and use the tower rule: E[	ext{revenue}] = E\big[\,E[	ext{revenue} \mid N]\,\big]. Step 1, freeze the count. If N = n customers come, each spending \12 on average, E[	ext{revenue} \mid N = n] = 12\,n. Step 2, average over the count. Only the mean of N is needed here, and it is given as E[N] = 40. So E[	ext{revenue}] = E[12\,N] = 12\,E[N] = 12 	imes 40 = 480. You expect \480 in revenue: expected number of customers (40) times average sp…

A random number of customers, each an average spend

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