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Chapter 4: Q.4.38 (page 166)
If E[X] = 1 and Var(X) = 5, find
(a) E[(2 + X)2];
(b) Var(4 + 3X).
In the given information the answers are (a)
(b)
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The random variable X is said to have the Yule-Simons distribution if
(a) Show that the preceding is actually a probability mass function. That is, show that
(b) Show that E[X] = 2.
(c) Show that E[X2] = q
Consider Problem 4.22 with i = 2. Find the variance of the number of games played, and show that this number is maximized when p = 1 2 .
There are two possible causes for a breakdown of a machine. To check the first possibility would cost C1 dollars, and, if that were the cause of the breakdown, the trouble could be repaired at a cost of R1 dollars. Similarly, there are costs C2 and R2 associated with the second possibility. Let p and 1 − p denote, respectively, the probabilities that the breakdown is caused by the first and second possibilities. Under what conditions on p, Ci, Ri, i = 1, 2, should we check the first possible cause of breakdown and then the second, as opposed to reversing the checking order, so as to minimize the expected cost involved in returning the machine to working order?
Compare the Poisson approximation with the correct binomial probability for the following cases:
when
when
when
when
In some military courts, judges are appointed. However, both the prosecution and the defense attorneys are entitled to a peremptory challenge of any judge, in which case that judge is removed from the case and is not replaced. A defendant is declared guilty if the majority of judges cast votes of guilty, and he or she is declared innocent otherwise. Suppose that when the defendant is, in fact, guilty, each judge will (independently) vote guilty with probability .whereas when the defendant is, in fact, innocent, this probability drops to .
(a) What is the probability that a guilty defendant is declared guilty when there are (i) , (ii) , and (iii) judges?
(b) Repeat part (a) for an innocent defendant.
(c) If the prosecuting attorney does not exercise the right to a peremptory challenge of a judge, and if the defense is limited to at most two such challenges, how many challenges should the defense attorney make if he or she is percent certain that the client is guilty?
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