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 flour 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 flour per week is given by Y = 3 X 2 +1. a. Find the median of X . b. Find the 20 th percentile of the distribution of X . c. What is the variance of X ? d. Find 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 o ff spring are both 0.50, find the proba- bility that a family’s fifth child is their second son. b. Suppose the probability that a car will have a flat 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 flat 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 e ff ects. What is the probability that the tenth patient will be the first to show these side e ff ects. d. Suppose X follows the geometric probability distribution with p = 0 . 2. Find P ( X 10). e. Let X b ( n, 0 . 4). Find 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 fixed 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, find 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 o ce all are busy 60% of the time. a. If you are calling this o ce, what is the probability that you complete your call on the first try? the second try? the third try? b. If you and your friend must both complete calls to this o ce, 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 profit per foot when the rope is sold is given by Y , where Y = 50 2 X X 2 . Find the expected profit per foot.
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