review_discrete - Review problems Problem 1 Suppose X b(n p...

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Review problems Problem 1 Suppose X b ( n, p ). a. The calculation of binomial probabilities can be computed by means of the following recursion formula . Verify this formula. P ( X = x +1)= p ( n x ) ( x +1)(1 p ) P ( X = x ) b. Let X b (8 , 0 . 25). Use the above result to calculate P ( X = 1), and P ( X =2) . You are given that P ( X =0)=0 . 1001 . Problem 2 The amount of Four used per week by a bakery is a random variable X having an exponential distribution with mean equal to 4 tons. The cost of the Four per week is given by Y =3 X 2 +1. a. ±ind the median of X . b. ±ind the 20 th percentile of the distribution of X . c. What is the variance of X ? d. ±ind P ( X> 6 /X > 2). e. What is the expected cost? Problem 3 Answer the folowing questions: a. If the probabilities of having a male or female o²spring are both 0.50, ³nd the proba- bility that a family’s ³fth child is their second son. b. Suppose the probability that a car will have a Fat tire while driving on the 405 freeway is 0.0004. What is the probability that of 10000 cars driving on the 405 freeway fewer than 3 will have a Fat tire. Use the Poisson approximation to binomial for faster calculations. c. A doctor knows from experience that 15% of the patients who are given a certain medicine will have udesirable side e²ects. What is the probability that the tenth patient will be the ³rst to show these side e²ects. d. Suppose X follows the geometric probability distribution with p =0 . 2. ±ind P ( X 10). e. Let X b ( n, 0 . 4). ±ind n so that P ( X 1) = 0 . 99. 11
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Problem 4 For a certain section of a pine forest, the number of diseased trees per acre, X , follows the Poisson distribution with λ = 10. The diseased trees are sprayed with an insecticide at a cost of $3 . 00 per tree, plus a ±xed overhead cost for equipment rental of $50 . 00. a. Find the probability that a randomly selected acre from this forest will contain at least 12 diseased trees. b. Letting C denote the total cost for a randomly selected acre, ±nd the expected value and standard deviation of C . Problem 5 A particular sale involves 4 items randomly selected from a large lot that is known to contain 10% defectives. Let X denote the number of defectives among the 4 sold. The purchaser of the items will return the defectives for repair, and the repair cost is given by C =3 X 2 + X +2. Find the expected repair cost. Problem 6 The telephone lines serving an airline reservation office all are busy 60% of the time. a. If you are calling this office, what is the probability that you complete your call on the ±rst try? the second try? the third try? b. If you and your friend must both complete calls to this office, what is the probability that it takes a total of 4 tries for both of you to get through? Problem 7 In the daily production of a certain kind of rope, the number of defects per foot X is assumed to have a Poisson distribution with mean λ = 2. The pro±t per foot when the rope is sold is given by Y , where Y =50 2 X X 2 . Find the expected pro±t per foot.
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review_discrete - Review problems Problem 1 Suppose X b(n p...

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