Chapter 5: Q. 5.18 (page 213)
Suppose that X is a normal random variable with
mean 5. If P{X > 9} = .2, approximately what is Var(X)?
The required variance is.
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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.)
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.
Compute the hazard rate function of a Weibull random variable and show it is increasing when and decreasing when
Consider Example 4b of Chapter 4, but now suppose that the seasonal demand is a continuous random variable having probability density function . Show that the optimal amount to stock is the value that satisfies
where is net profit per unit sale, is the net loss per unit
unsold, and is the cumulative distribution function of the
seasonal demand.
Your company must make a sealed bid for a construction project. If you succeed in winning the contract (by having the lowest bid), then you plan to pay another firm $100,000 to do the work. If you believe that the minimum bid (in thousands of dollars) of the other participating companies can be modeled as the value of a random variable that is uniformly distributed on (70, 140), how much should you bid to maximize your expected profit?
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