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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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A fair die is rolled times. Calculate the expected sum of the rolls.
Show that if and are independent, then
(a) in the discrete case;
(b) in the continuous case.
A bottle initially contains m large pills and n small pills. Each day, a patient randomly chooses one of the pills. If a small pill is chosen, then that pill is eaten. If a large pill is chosen, then the pill is broken in two; one part is returned to the bottle (and is now considered a small pill) and the other part is then eaten.
(a) Let X denote the number of small pills in the bottle after the last large pill has been chosen and its smaller half returned. Find E[X].
Hint: De铿乶e n + m indicator variables, one for each of the small pills initially present and one for each of the small pills created when a large one is split in two. Now use the argument of Example m.
(b) Let Y denote the day on which the last large pills chosen. Find E[Y].
Hint: What is the relationship between X and Y?
Gambles are independent, and each one results in the player being equally likely to win or lose 1 unit. Let W denote the net winnings of a gambler whose strategy is to stop gambling immediately after his first win. Find
(a) P{W > 0}
(b) P{W < 0}
(c) E[W]
Suppose that A and B each randomly and independently chooseofobjects. Find the expected number of objects
a. Chosen by both A and B;
b. Not chosen by either A or B;
c. Chosen by exactly one of A and B.
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