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Chapter 7: Q. 7.5 (page 359)
Let be arbitrary events, and define
{at least of the occur}. Show that
Hint: Let denote the number of the that occur. Show
that both sides of the preceding equation are equal to .
The arbitrary events is showed as.
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A population is made up of disjoint subgroups. Let denote the proportion of the population that is in subgroup . If the average weight of the members of subgroup is , what is the average weight of the members of the population?
Consider the following dice game: A pair of dice is rolled. If the sum isthen the game ends and you win If the sum is not then you have the option of either stopping the game and receiving an amount equal to that sum or starting over again. For each value of find your expected return if you employ the strategy of stopping the first time that a value at least as large as appears. What value ofleads to the largest expected return? Hint: Let denote the return when you use the critical value To compute, condition on the initial sum.
If where a and b are constants, express the moment generating function of in terms of the moment generating function of .
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.
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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