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Chapter 7: Q.23 (page 360)
Show that is a standard normal random variable and if is defined by , then
We prove that
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Let Z be a standard normal random variable,and, for a 铿亁ed x, set
The number of accidents that a person has in a given year is a Poisson random variable with mean 蹋 However, suppose that the value of changes from person to person, being equal to for percent of the population and for the other percent. If a person is chosen at random, what is the probability that he will have
(a) accidents and,
(b) Exactly accidents in a certain year? What is the conditional probability that he will have accidents in a given year, given that he had no accidents the preceding year?
Show that is stochastically larger than if and only if
for all increasing functions .
Hint: Show that , then by showing that and then using Theoretical Exercise 7.7. To show that if for all increasing functions , then , define an appropriate increasing function .
The joint density of and is given by
,
(a) Compute the joint moment generating function of and .
(b) Compute the individual moment generating functions.
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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