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Chapter 4: Q.4.8 (page 170)
Let be a random variable having expected value and variance . Find the expected value and variance of.
Mean is , and variance is .
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Two coins are to be 铿俰pped. The 铿乺st coin will land on heads with probability ., the second with probability .. Assume that the results of the 铿俰ps are independent, and let X equal the total number of heads that result. (a) Find P{X =}. (b) Determine E[X].
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.
The expected number of typographical errors on a page of a certain magazine is. What are the probability that the next page you read contains (a) and (b) or more typographical errors? Explain your reasoning!
Letbe the winnings of a gambler. Let and suppose that
Compute the conditional probability that the gambler wins given that he wins a positive amount.
Let be a negative binomial random variable with parameters and , and let be a binomial random variable with parameters and . Show that
Hint: Either one could attempt an analytical proof of the preceding equation, which is equivalent to proving the identity
or one could attempt a proof that uses the probabilistic interpretation of these random variables. That is, in the latter case, start by considering a sequence of independent trials having a common probability p of success. Then try to express the events to express the events and in terms of the outcomes of this sequence.
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