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Chapter 5: Q.5.6 (page 212)
5.6. Computeif has a density function given by
;
;
.
Thus the answer of given question is
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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)
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.)
Suppose that the height, in inches, of a -year-old man is a normal random variable with parameters role="math" localid="1646741074533" . What percentage of -year-old men are taller than feet, inches? What percentage of men in the -footer club are taller than feet, inches?
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.
The lifetimes of interactive computer chips produced
by a certain semiconductor manufacturer are normally distributed with parametershours and hours. What is the approximate probability that abatch of chips will contain at least whose lifetimes are less than ?
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