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Chapter 5: Q 5.37 (page 214)
Ifis uniformly distributed over find
the density function of the random variable.
Therefore,
We have found that
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Let X and Y be independent random variables that are both equally likely to be either 1, 2, . . . ,(10)N, where N is very large. Let D denote the greatest common divisor of X and Y, and let Q k = P{D = k}.
(a) Give a heuristic argument that Q k = 1 k2 Q1. Hint: Note that in order for D to equal k, k must divide both X and Y and also X/k, and Y/k must be relatively prime. (That is, X/k, and Y/k must have a greatest common divisor equal to 1.) (b) Use part (a) to show that Q1 = P{X and Y are relatively prime} = 1 q k=1 1/k2 It is a well-known identity that !q 1 1/k2 = π2/6, so Q1 = 6/π2. (In number theory, this is known as the Legendre theorem.) (c) Now argue that Q1 = "q i=1 P2 i − 1 P2 i where Pi is the smallest prime greater than 1. Hint: X and Y will be relatively prime if they have no common prime factors. Hence, from part (b), we see that Problem 11 of Chapter 4 is that X and Y are relatively prime if XY has no multiple prime factors.)
Every day Jo practices her tennis serve by continually serving until she has had a total of successful serves. If each of her serves is, independently of previous ones,
successful with probability , approximately what is the probability that she will need more than serves to accomplish her goal?
Hint: Imagine even if Jo is successful that she continues to serve until she has served exactly times. What must be true about her first serves if she is to reach her goal?
The life of a certain type of automobile tire is normally distributed with mean miles and standard deviation miles.
(a) What is the probability that such a tire lasts more than miles?
(b) What is the probability that it lasts between andmiles?
(c) Given that it has survived miles, what is the conditional probability that the tire survives another miles?
A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you bet 1 on a specified number, then you either win 35 if the roulette ball lands on that number or lose 1 if it does not. If you continually make such bets, approximate the probability that
(a) you are winning after 34 bets;
(b) you are winning after 1000 bets;
(c) you are winning after 100,000 bets
Assume that each roll of the roulette ball is equally likely to land on any of the 38 numbers
You arrive at a bus stop at a.m., knowing that the bus will arrive at some time uniformly distributed between and .
(a) What is the probability that you will have to wait longer than minutes?
(b) If, at , the bus has not yet arrived, what is the probability that you will have to wait at least an additional minutes?
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