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Chapter 4: Q.4.20 (page 174)
Show that if X is a geometric random variable with parameter p, then Hint: You will need to evaluate an expression of the form
to do so, write
and then interchange the sum and the integral.In the given information the answer is
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The National Basketball Association (NBA) draft lottery involves the 11 teams that had the worst won-lost records during the year. A total of 66 balls are placed in an urn. Each of these balls is inscribed with the name of a team: Eleven have the name of the team with the worst record, 10 have the name of the team with the second worst record, 9 have the name of the team with the third worst record, and so on (with 1 ball having the name of the team with the 11 th-worst record). A ball is then chosen at random, and the team whose name is on the ball is given the first pick in the draft of players about to enter the league. Another ball is then chosen, and if it "belongs" to a team different from the one that received the first draft pick, then the team to which it belongs receives the second draft pick. (If the ball belongs to the team receiving the first pick, then it is discarded and another one is chosen; this continues until the ball of another team is chosen.) Finally, another ball is chosen, and the team named on the ball (provided that it is different from the previous two teams) receives the third draft pick. The remaining draft picks 4 through 11 are then awarded to the 8 teams that did not "win the lottery," in inverse order of their won-lost not receive any of the 3 lottery picks, then that team would receive the fourth draft pick. Let X denote the draft pick of the team with the worst record. Find the probability mass function of X.
A newsboy purchases papers at cents and sells them at cents. However, he is not allowed to return unsold papers. If his daily demand is a binomial random variable with , approximately how many papers should he purchase so as to maximize his expected profit?
Three dice are rolled. By assuming that each of the possible outcomes is equally likely, find the probabilities attached to the possible values that X can take on, where X is the sum of the 3 dice.
and will take the same -question examination. Each question will be answered correctly by with probability, independently of her results on other questions. Each question will be answered correctly by B with probability , independently both of her results on the other questions and on the performance of
(a) Find the expected number of questions that are answered correctly by both A and B.(b) Find the variance of the number of questions that are answered correctly by either A or B
Consider a random collection of individuals. In approximating the probability that no of these individuals share the same birthday, a better Poisson approximation than that obtained in the text (at least for values of between and ) is obtained by letting be the event that there are at least 3 birthdays on day
(a) Find .
(b) Give an approximation for the probability that noindividuals share the same birthday.
(c) Evaluate the preceding when (which can be shown to be the smallest value offor which the probability exceeds.).
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