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Chapter 5: Q. 5.11 (page 212)
A point is chosen at random on a line segment of
length . Interpret this statement, and find the probability
that the ratio of the shorter to the longer segment is
less than .
The required probability is.
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A bus travels between the two cities A and B, which are miles apart. If the bus has a breakdown, the distance from the breakdown to city A has a uniform distribution over . There is a bus service station in city A, in B, and in the center of the route between A and B. It is suggested that it would be more efficient to have the three stations located miles, respectively, from A. Do you agree? Why?
The annual rainfall (in inches) in a certain region is normally distributed with . What is the probability that starting with this year, it will take more than years before a year occurs having a rainfall of more than inches? What assumptions are you making?
The mode of a continuous random variable having density is the value of for which attains its maximum. Compute the mode of in cases and of Theoretical Exercise
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.
If is uniformly distributed over , what random variable, having a linear relation with , is uniformly distributed over
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