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Chapter 5: Q.5.18 (page 213)
Suppose that X is a normal random variable with mean . If , approximately what is ?
The required variance is 22.66.
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The median of a continuous random variable having distribution function F is that value m such that F(m) = . That is, a random variable is just as likely to be larger than its median as it is to be smaller. Find the median of X if X is
(a) uniformly distributed over (a, b);
(b) normal with parameters 渭,蟽;
(c) exponential with rate 位.
A standard Cauchy random variable has density function
Show that if X is a standard Cauchy random variable, then 1/X is also a standard Cauchy random variable.
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.
(a) A fire station is to be located along a road of length . If fires occur at points uniformly chosen on localid="1646880402145" , where should the station be located so as to minimize the expected distance from the fire? That is,
choose a so as to minimize localid="1646880570154" when X is uniformly distributed over .
(b) Now suppose that the road is of infinite length鈥� stretching from point outward to . If the distance of a fire from point is exponentially distributed with rate , where should the fire station now be located? That is, we want to minimize , where X is now exponential with rate .
The number of years that a washing machine functions is a random variable whose hazard rate function is given by
(a)What is the probability that the machine will still be working years after being purchased?
(b) If it is still working years after being purchased, what is the conditional probability that it will fail within the next
years?
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