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Chapter 7: Q. 7.51 (page 363)
Use Table to determine the distribution of when are independent and identically distributed exponential random variables, each having mean.
The identically distributed exponential random variables value are.
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For an event A, let IA equal 1 if A occurs and let it equal 0 if A does not occur. For a random variable X, show that E[X|A] = E[XIA] P(A
In the text, we noted that
when the are all nonnegative random variables. Since
an integral is a limit of sums, one might expect that
whenever are all nonnegative random
variables; this result is indeed true. Use it to give another proof of the result that for a nonnegative random variable ,
Hint: Define, for each nonnegative , the random variable
by
role="math" localid="1647348183162"
Now relate
The random variables X and Y have a joint density function is given by
Compute
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?
For Example , show that the variance of the number of coupons needed to a mass a full set is equal to
When is large, this can be shown to be approximately equal (in the sense that their ratio approaches 1 as ) to .
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