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Chapter 5: Q.5.11 (page 215)
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].
Showing that ,
a)
b)
c)
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Suppose that the travel time from your home to your office is normally distributed with mean minutes and standard deviation minutes. If you want to be percent certain that you will not be late for an office appointment at p.m., what is the latest time that you should leave home?
A model for the movement of a stock supposes that if the present price of the stock is , then after one period, it will be either with probability or with probability . Assuming that successive movements are independent, approximate the probability that the stock鈥檚 price will be up at least percent after the next periods if
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.Use the result that for a nonnegative random variable,
to show that for a nonnegative random variable,
Hint: Start with
and make the change of variables.
The random variable has the probability density function
If , find
(a) and
(b) .
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