Question

2. The number of customers entering Heal's in a given hour is Poisson distributed with mean 30. The amount of money (in pounds) spent by each customer is uniformly distributed over (0, 500). Let T denote the total amount of money spent by customers in Heal's in one hour. Find ET). You should define any notation you use and note any assumptions you make. HINT 1: Write T as a sum of random variables, noting that the number of random variables to be summed is also random. HINT 2: Try using iterated conditional expectation to find this expectation - what variable would be sensible to condition on? Would conditioning on the number of people to enter Heal's in one hour be a sensible choice?

Answer 1

let y be the number of customers entering Heal's xr in a given your Poisson (30) (as E (x) = and here = 30 so xn Poisson ( Lett - Amount of money spent by each custom Y r uniform (0, 500) sox E(T) / ECE (*XXX Given T = total amount of money spent by customers in Heal's in one hour, Y; " is the amount spent l by each customers and in total there are x customers) flere x is independent of each sandom variable Yo and X can lake values 0, 1, 2, 3 .. E (T1x = x) = E(81) E1%) 2 (5040) VIX = 850x = 500 to [W E14) 530) = 250

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