Answer to Question #120923 in Statistics and Probability for gwein Juspain

Question

Suppose that the number of people entering a department store on a given day is a
random variable with mean 50. Suppose further that the amounts of money spent by
these customers are independent random variables having a common mean of GHS
12.00. Finally, suppose also that the amount of money spent by a customer is also
independent of the total number of customers who enter the store. What is the expected
amount of money spent in the store on a given day?

Answer 1

Let X = the random variable denoting the number of people entering a department store on a given day

and Y = the random variable denoting the amounts of money spent by a customer

Therefore, XY = the random variable denoting the amount of money spent by all the customers

We are given, E(X) = 50 and E(Y) = 12.00

The amount of money spent by a customer is independent of the total number of customers who enter the store, i.e. X and Y are independent random variables.

$\therefore$ The expected amount of money spent in the store on a given day is

E(XY)

= E(X).E(Y) [since X and Y are independent random variables]

= 50 $\times$ 12.00 = 600.00

Answer: The expected amount of money spent in the store on a given day is GHS 600.00.