P(X = x k ) = 1 = k=1
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1 74 CHAPTER 6. IMPORTANT DISTRIBUTIONS AND DENSITIES 6.2 Problems Which are modeled with a unifm distribution? (a Yes, P(X k 1/6 f k 1,...,6. (b No, this has a binomial distribution. (c Yes, P(X k 1/38 f k 0, 00, 1,..., 36. (d Yes (e No, geometric Show that a random variable X which can take on a countably infinite set of values cannot be unifmly distributed. Suppose the outcomes are x k f k 1, 2, 3,.... If X were unifmly distributed, then we would have P(X x k p f some number p independent of k. A probability distribution also has the requirement that 1 P(X x k k1 p. k1 If p > 0 the sum is infinite, not 1. If p 0 the sum is 0. Thus there is no value of p with 0 p 1 that can be used Let X 1, X 2,...,X N be N mutually independent random variables, each of which is unifmly distributed on the integers from 1 to K. Let Y denote the minimum of the X n s. Find the distribution of Y. The possible values f Y are 1,...,K. In addition, P(X n k 1/K. Writing the event {Y j} as the union of disjoint events it follows that {Y j} {Y j} {Y j + 1}, P(Y j P(Y j + P(Y j + 1, P(Y j P(Y j P(Y j + 1. Now the probability that the minimum of the X n is at least j is the same as the probability that all of them are at least j, so by independence, P(Y j P({X 1 j} {X N j} P({X 1 j} P({X N j} (K + 1 j N /K N, j 1,...,K.
2 6.2. PROBLEMS 75 Thus P(Y j (K + 1 j N /K N (K j N /K N, j 1,..., K, with P(Y j 0 otherwise A die is rolled until the first time T that a 6 turns up. (a What is the probability distribution f T? (b Find P(T > 3. (c Find P(T > 6 T > 3 T has a geometric distribution with p 1/6. Let q 1 p 5/6. Thus P(T > 3 q k p q 3 (5/6 3, k3 and by the memyless property of the geometric distribution (page 186 we have P(T > 6 T > 3 P(T > 3, which has just been given If a coin is tossed a sequence of times, what is the probability P that the first head will occur after the fifth toss, given that it has not occurred in the first two tosses? Let T be the toss when a heads first appears. T has the geometric distribution, with p 1/2. Then by the memyless property of the geometric distribution, P P(T > 5 T > 2 P(T > 3 (1/2 3 1/ Estimate the number of trout by catching, tagging them, catching another, finding of the second group tagged. (a Give a rough estimate of the number of trout in the lake. (b Let N be the number of trout in the lake. Find an expression f the probability of getting tagged trout. (c Find the value of N maximizing the probability in (b. (a The total number should be about 0, since the tagged ones appear to be about ten percent of the total. (b The probability of catching tagged fish is P N ( ( N /.
3 76 CHAPTER 6. IMPORTANT DISTRIBUTIONS AND DENSITIES (c Following the hint, we consider the ratio of successive values of N. We find P N+1 /P N ( ( (N 99!N!(N 99!(N 1! (N 189!(N!(N + 1!(N! (N 99 2 (N 189(N + 1 We want the biggest value of N such that P N+1 /P N After some simplification we get (N 99 2 (N 189(N + 1 1, N , N 999. The biggest values of P N are achieved at N 999 N Let N be the number of people living in a certain area. The census counts n 1 people. They return to the same area f a recount, finding people the second time, of whom n 12 have been counted both times. (a Let X be the number counted both times. Find P(X k. (b Find the value of N which maximizes P(X n12. (a The first thing to notice is that this is essentially the same problem as fish tagging. The first count has tagged, painted red, n 1 of N people. We view the second count as a sample of out of N, and ask f the probability of finding k red ones, that is, people in the sample who were counted befe. This is given by the hypergeometric distribution h(n, n 1, ; k ( n1 k n1 k. (b As in problem 9, consider the ratio P N+1 /P N h(n + 1, n 1, ; k/h(n, n 1, ; k.
4 6.2. PROBLEMS 77 After preliminary cancellation, P N+1 /P N +1 n1 k +1 n1 k (N + 1 n 1! ( k!(n + 1 n 1 + k! N!!(N!!(N + 1! (N + 1! ( k!(n n 1 + k! (N n 1! (N + 1 n 1 (N + 1 (N + 1 n 1 + k (N + 1 To find the maximum of P N we want to find the biggest values of N with P N+1 /P N 1. That is, we want to solve (N + 1 n 1 (N + 1 (N + 1 n 1 + k(n + 1, (n 1 + (N + 1 n 1 (n 1 + k(n + 1. The biggest value of N is thus N n 1 k On average only 1 person in 0 has a particular rare blood type. (a Find the probability that, in a city of, 000, no one has this blood type. (b How many people would have to be tested to give a probability greater than 1/2 of finding at least one person with this blood type? (a P (.999, (b Let N be the number tested and let X be the number of people with the blood type. We approximate the distribution of X by a Poisson random variable with average number λ.001n. We want to find N so that P(X 1 > 1/2. Solve P(X 1 1 P(X 0 1 e λ 1 e.001n.5 We want e.001n.5, N 693.
5 78 CHAPTER 6. IMPORTANT DISTRIBUTIONS AND DENSITIES An operat receives on average one call every seconds. If the operat takes a 300 second break, what is the probability of missing at most one call? As suggested, use the Poisson approximation with λ 3. The probability of missing 0 1 calls is e e An advertiser drops, 000 leaflets on a city which has 2000 blocks. Assume that each leaflet has an equal chance of landing on each block. What is the probability that a particular block will receive no leaflets? Again use the Poisson approximation with λ /2 5. The probability of a block getting no leaflets is approximately e
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