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Fall 2018 S&DS 241: Probability Theory with Applications Homework 5 Due: Oct 10, 2018 in class Prof. Yihong Wu 1. (20 pts) Blitzstein-Hwang, Chapter 4, Problem 67 2. (20 pts) Consider the coupon collector problem discussed in the lecture. There are n different types of coupons. Each box of cereal contains one coupon whose type is chosen independently and equally likely to be any of the n types. A coupon collector buys k boxes, where k ≥ n. (a) Show that the probability that the first coupon is not collected is (1 −

1 k n) .

k  (b) Show that the probability that all coupons are collected is at least 1 − n 1 − n1 . (Hint: union bound) (c) Using part (b), give an estimate on how many boxes to buy in order to guarantee the probability to complete the collection is at least 99%. Evaluate your estimate for n = 20.

3. (20 pts) Compound Poisson. Let λ > 0 be a fixed real number. Consider the following experiment: generate a random variable X according to the Poi(λ) distribution. Then flip a fair coin for X times. Let Y denote the number of heads and Z the number of tails. (a) Show that both Y and Z are distributed according to Poi(λ/2). (b) Show that Y and Z are independent. 4. (20 pts) Debbie takes a flight from SFO to HVN with a layover at PHL. She has one piece of checked luggage. At the SFO airport there is a probability of 1/2 that her luggage is lost. This probability is 1/3 at PHL and 1/4 at HVN. Assume what happened at the three airports are mutually independent. (a) What is the probability that Debbie’s luggage is lost? (b) It turns out that Debbie’s luggage was lost. Conditioned on this, what is the probability that it was lost at PHL? (c) Suppose that Debbie’s luggage was recovered. SFO takes 5 days to deliver any lost luggage, PHL takes 3 days, and HVN takes 1 day. What is the expected number of days to wait for her luggage to arrive? 5. (20 pts) Bold play. For the gambler’s ruin problem, in the lecture we discussed the version of timid play, where in each game, the gambler makes a small bet of fixed amount, until he/she is ruined or reaches a given target fortune. Next, let’s discuss bold play, where in each game, the gambler bets either his/her entire fortune or the amount needed to reach the target fortune, whichever is smaller. Suppose the gambler, who is bold, starts with $x, where x ∈ [0, 1], and the target fortune is $1. Suppose each bet is won with probability p or lost with probability q = 1 − p, independently. Let F (x) denotes the winning probability, i.e., the probability that the bold gambler reaches the target fortune. (a) Show that F (0) = 0 and F (1) = 1. 1

(b) Show that F satisfies the following relation: ( p · F (2x) 0 ≤ x ≤ 12 F (x) = p + q · F (2x − 1) 21 ≤ x ≤ 1 (c) Show that   1 =p 2   3 = p + pq F 4   5 F = p + p2 q 8

F

(d) Show that   p2 1 . = F 1 − pq 3 (e) (Optional) Let’s compare the winning chance with that of timid play. Consider the initial fortune x = 34 . A timid gambler bets $ 14 each game. Plot the winning probability of a timid gambler and that of a bold gambler as a function of p. Compare two curves. It turns out that for unfavorable games (p < 21), bold play is always better, hence the saying desperate times call for desperate measures. Does this sound counterintuitive to you?

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