Question

4.8. Each box of a certain brand of cereal comes with a toy inside. If there are n possible toys and if the toys are distributed randomly, how many boxes do you expect to buy before you get them all? Parts (a) and (b) describe a way of getting an answer. (a) Assuming that you already have ℓ of the toys, let Xℓ be the number of boxes you need to purchase until you get a new toy that you don’t already have. Compute the expected value ErXℓs. [Hint: We can think of each new box purchased as a “coin flip” where H ““we get a new toy” and T ““we don’t get a new toy.” Thus Xℓ is a geometric random variable. What is PpHq? See also Exercise 3.1.] (b) Let X be the number of boxes you purchase until you get all n toys. Thus we have X “ X0 ` X1 ` X2 ` ¨¨¨ ` Xn´1. Use part (a) and linearity to compute the expected value ErXs. [Note: You may leave your answer as a summation.] (c) Application: Suppose you continue to roll a fair 6-sided die until you see all six sides. How many rolls do you expect to make? (a) Assuming that you already have ℓ of the toys, let Xℓ be the number of boxes you need to purchase until you get a new toy that you don’t already have. Compute the expected value ErXℓs. [Hint: We can think of each new box purchased as a “coin flip” where

Answer 1