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Chapter 5: Q 5.39 (page 214)
If is an exponential random variable with a parameter, compute the probability density function of the random variable defined by
Therefore, the probability density function of the random variable
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Find the distribution of, where is a fixed constant and is uniformly distributed on. Such a random variable arises in the theory of ballistics. If a projectile is fired from the origin at an angle from the earth with a speed, then the point at which it returns to the earth can be expressed as, where is the gravitational constant, equal to centimeters per second squared.
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.)
The random variable X is said to be a discrete uniform random variable on the integers 1, 2, . . . , n if P{X = i } = 1 n i = 1, 2, . . , n For any nonnegative real number x, let In t(x) (sometimes written as [x]) be the largest integer that is less than or equal to x. Show that if U is a uniform random variable on (0, 1), then X = In t (n U) + 1 is a discrete uniform random variable on 1, . . . , n.
The number of minutes of playing time of a certain high school basketball player in a randomly chosen game is a random variable whose probability density function is given in the following figure:
Find the probability that the player plays
(a) more than minutes;
(b) between minutes;
(c) less than minutes;
(d) more than minutes
A system consisting of one original unit plus a spare
can function for a random amount of time. If the density
ofis given (in units of months) by
what is the probability that the system functions for at least months?
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