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Chapter 5: Q. 5.15 (page 213)
If is a normal random variable with parameters and , compute
(a)role="math" localid="1646719347104"
(b)role="math" localid="1646719357568"
(c)role="math" localid="1646719367217"
(d)
(e)
The required answers are:
(a)
(b)
(c)role="math" localid="1646720604668"
(d)
(e)
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Verify thatwhenis a gamma random variable with parameters and
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 = 蟺2/6, so Q1 = 6/蟺2. (In number theory, this is known as the Legendre theorem.) (c) Now argue that Q1 = "q i=1 P2 i 鈭� 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.)
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 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 , how much should you bid to maximize your expected profit?
Let X have probability density f X. Find the probability density function of the random variable Y defined by Y = a X + b.
A filling station is supplied with gasoline once a week. If its weekly volume of sales in thousands of gallons is a random variable with probability density function
what must the capacity of the tank be so that the probability of the supply being exhausted in a given week is role="math" localid="1646634562935"
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