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Chapter 5: Q. 5.19.TE (page 216)
If is an exponential random variable with a mean , show that
The statement has been proved true, i.e.
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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?
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 random variable has the probability density function
If , find
(a) and
(b) .
A fire station is to be located along a road of length. If fires occur at points uniformly chosen on, where should the station be located so as to minimize the expected distance from the fire? That is, choose a so as to
minimize
whenis uniformly distributed over
Now suppose that the road is of infinite length鈥� stretching from point outward to. If the distance of fire from the point is exponentially distributed with rate, where should the fire station now be located? That is, we want to minimize, where is now exponential with rate.Suppose that the cumulative distribution function of the random variable is given by
Evaluate ; (c) the hazard rate function of .
Hint: For parts and , you might want to make use of the results of Theoretical Exercise .
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