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Chapter 7: Q. 7.48 (page 363)
If where a and b are constants, express the moment generating function of in terms of the moment generating function of .
The moment generating function and in the fourth equality we used linearity of expectation.
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7.2. Suppose that is a continuous random variable with
density function . Show that is minimized
when is equal to the median of .
Hint: Write
Now break up the integral into the regions where
and where , and differentiate.
A prisoner is trapped in a cell containingdoors. The first door leads to a tunnel that returns him to his cell after days’ travel. The second leads to a tunnel that returns him to his cell after days’ travel. The third door leads to freedom after day of travel. If it is assumed that the prisoner will always select doors and with respective probabilities and ., what is the expected number of days until the prisoner reaches freedom?
The number of people who enter an elevator on the ground floor is a Poisson random variable with mean . If there are floors above the ground floor, and if each person is equally likely to get off at any one of the floors, independently of where the others get off, compute the expected number of stops that the elevator will make before discharging all of its passengers.
An urn contains balls, of whichare red and 8 are blue. From this urn, 12 balls are randomly withdrawn. Let X denote the number of red and Y the number of blue balls that are withdrawn. Find Cov(X, Y)
(a) by defining appropriate indicator (that is, Bernoulli) random variables
such that
(b) by conditioning (on either X or Y) to determine
Urn contains white and black balls, while urn contains white and black balls. Two balls are randomly selected from urn and are put into urn . If balls are then randomly selected from urn , compute the expected number of white balls in the trio.
Hint: LetXi = if the i th white ball initially in urn is one of the three selected, and let Xi = otherwise. Similarly, let Yi = if the i the white ball from urn is one of the three selected, and let Yi = otherwise. The number of white balls in the trio can now be written as
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