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Chapter 5: Q. 5.14 (page 213)
Let be a uniform random variable. Compute role="math" localid="1646717640777" by using Proposition , and then check the result by using the definition of expectation.
The required answer is.
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Find the probability density function of Y = eX when X is normally distributed with parameters μ and σ2. The random variable Y is said to have a lognormal distribution (since log Y has a normal distribution) with parameters μ and σ2.
The salaries of physicians in a certain specialty are approximately normally distributed. If percent of these
physicians earn less than and percent earn more than , approximately what fraction earn
(a) less than ?
(b) between ?
Each item produced by a certain manufacturer is, independently, of acceptable quality with probability . Approximate the probability that at most of the next items produced are unacceptable.
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.)
Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked is an exponential random variable with a parameter. Smith has a used car that he claims has been driven only miles. If Jones purchases the car, what is the
probability that she would get at least additional miles out of it? Repeat under the assumption that the life-
time mileage of the car is not exponentially distributed, but rather is (in thousands of miles) uniformly distributed over.
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