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Chapter 4: Q.4.75 (page 168)
A fair coin is continually flipped until heads appears for the 10th time. Let X denote the number of tails that occur. Compute the probability mass function of X.
In the given information the answer is.
this distribution is well known as negative binomial distribution
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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?
Let be a Poisson random variable with parameter . What value of maximizes
An urn has n white and m black balls. Balls are randomly withdrawn, without replacement, until a total of white balls have been withdrawn. The random variable equal to the total number of balls that are withdrawn is said to be a negative hypergeometric random variable.
(a) Explain how such a random variable differs from a negative binomial random variable.
(b) Find .
Hint for (b): In order for to happen, what must be the results of the first withdrawals?
A communications channel transmits the digits and However, due to static, the digit transmitted is incorrectly received with probability Suppose that we want to transmit an important message consisting of one binary digit. To reduce the chance of error, we transmit instead of and 11111 instead of If the receiver of the message uses “majority” decoding, what is the probability that the message will be wrong when decoded? What independence assumptions are you making?
A certain typing agency employs typists. The average number of errors per article is when typed by the first typist and when typed by the second. If your article is equally likely to be typed by either typist, approximate the probability that it will have no errors.
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