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Chapter 5: Q. 5.18 (page 213)
Suppose that X is a normal random variable with
mean 5. If P{X > 9} = .2, approximately what is Var(X)?
The required variance is.
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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.
Show that
Hint: Make the change of variables and then relate the resulting expression to the normal distribution.
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.
Let Z be a standard normal random variable Z, and let g be a differentiable function with derivative g'.
(a) Show that E[g'(Z)]=E[Zg(Z)];
(b) Show that E[Zn+]=nE[Zn-].
(c) Find E[Z].
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.What do you think about this solution?
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