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p92_exercises-3_7

p92_exercises-3_7 - 92 EXAMPLE 3—25 Web Servers servers...

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Unformatted text preview: 92 EXAMPLE 3—25 Web Servers servers? Consequently, a,mswt.wwsegm . . . . . EXERCISES FOR SECTION 3—7 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS E(X) = 3/0.0005 = 6000 requests P(XS 5) = P(X: 3) + P(X: 4) + P(X: 5) 3 4 = 0.00053 + (2) 0.00053(o.9995) + (2) 0.00053(0.9995)2 = 1.25 x 10—10 + 3.75 x 10‘10 + 7.49 x 10—10 = 1.249 x 10'9 M‘— 3—81. Suppose the random variableX has a geometric distribu- tion with p = 0.5. Determine the following probabilities: (a) P(X = 1) (b) P(X = 4) (c) P(X = 8) (d) P(X S 2) (e) P(X > 2) 3—82. Suppose the random variable Xhas a geometric distri- bution with a mean of 2.5. Determine the following probabilities: (a) P(X = 1) (b) P(X = 4) (c) P(X: 5) (d) P(Xs 3) (e) P(X > 3) 3—83. Consider a sequence of independent Bernoulli trials with p = 0.2. (a) What is the expected number of trials to obtain the ﬁrst success? (b) After the eighth success occurs, what is the expected num- ber of trials to obtain the ninth success? 3—84. Suppose thatX is a negative binomial random variable with p = 0.2 and r = 4. Determine the following: (a) E(X) (b) P(X= 20) (c) P(X = 19) (d) P(X = 21) (e) The most likely value forX 3—85. The probability of a successful optical alignment in the assembly of an optical data storage product is 0.8. Assume the trials are independent. (a) What is the probability that the ﬁrst successful alignment requires exactly four trials? (b) What is the probability that the ﬁrst successful alignment requires at most four trials? (c) What is the probability that the ﬁrst successﬁil alignment requires at least four trials? 3—86. In a clinical study, volunteers are tested for a gene that has been found to increase the risk for a disease. The probability that a person carries the gene is 0.1. (a) What is the probability 4 or more people will have to be tested before 2 with the gene are detected? (b) How many people are expected to be tested before 2 with the gene are detected? 3—87. Assume that each of your calls to a popular radio station has a probability of 0.02 of connecting, that is, of not obtaining a busy signal. Assume that your calls are independent. (a) What is the probability that your ﬁrst call that connects is your tenth call? (b) What is the probability that it requires more than ﬁve calls for you to connect? (c) What is the mean number of calls needed to connect? 3—88. A player of a video game is confronted with a series of I opponents and has an 80% probability of defeating each one. Success with any opponent is independent of previous encoun- ters. The player continues to contest opponents until defeated. (a) What is the probability mass ﬁmction of the number of opponents contested in a game? (b) What is the probability that a player defeats at least two opponents in a game? 4 i A Web site contains three identical computer servers. Only one is used to operate the site, and the other; two are spares that can be activated in case the primary system fails. The probability of a failure in the; primary computer (or any activated spare system) from a request for service is 0.0005. Assuming that; ' each request represents an independent trial, what is the mean number of requests until failure of all three ‘ Let X denote the number of requests until all three servers fail, and let X1, X2, and X3 denote the - ‘ number of requests before a failure of the ﬁrst, second, and third servers used, respectively. Now, . ‘ X = X1 + X2 + X3. Also, the requests are assumed to comprise independent trials with constant proba-j bility of failure p = 0.0005. Furthermore, a spare server is not affected by the number of requests before '4 . it is activated. Therefore, X has a negative binomial distribution with p = 0.0005 and r = 3. What is the probability that all three servers fail within ﬁve requests? The probability is P(X S 5) and ...
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