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au95final - Statistics 427 FINAL EXAM Autumn 795 Prof Prem...

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Unformatted text preview: Statistics 427 FINAL EXAM, Autumn 795 Prof. Prem Goel Answer all questions. Maximum score 100 points. The points for each question are given in [ ] Time limit : 1 hour 48 minutes. SHOW ALL YOUR WORK AND REASONING, IN ALL PARTS OF ALL QUESTIONS. IMPORTANT: If you can not do a part of a question and you need that answer for the subsequent parts assume any reasonable answer for the part you cannot do and proceed to get credit for the subsequent parts. DO NOT START writing until I ask you to, but you can read the questions in the mean time. If you have any question any time during this test, raise your hand. I will come to your seat to clarify. Good Luck! 1. Suppose that 10% of all flash bulbs manufactured by a certain company are defective. Let X be the number of defective bulbs in a random sample of 500 bars manufactured by that company. (a) What is the distribution of X. Specify the parameter(s). [3 points] (b) Find E(X) and Va7"(X). [7 points] (0) Find the probability (upto three decimal places) that there will be no more than 20 defectives in the sample. [10 points] . Let X1, . . . , X4 be a random sample from a distribution with probability density func- tion: 2 _ .75(1—:c) for —1£x§1, “33) _ { 0 elsewhere. (a) Find the distribution function F($) corresponding to the density function f (at) [10 points] (b) Let, X; be the minimum of X1, . . . 7X4. Find P(X; g .5). [10 points] . At the Universite de Montefoire, the number of hours spent in the library per student follows the pdf f(y) = k(6.0 — y), 0 < y < 6. (a) Find the value of the constant k so that the function f (y) given above is a pdf. [4 points] (b) Find the median number of hours spent by a randomly selected student in the library. [6 points] (0) Because of a limited capacity of the Study room, the library allows a student to use the Study room for at most 4 hours in a day. Find the probability that a student is forced to leave the study room on a given day. [10 points] (d) Each student needs approximately $1 for copying books and $0.50 for soda during each hour he/she spends in the library. Let X = 1.5 Y denote daily expenditure per student in the library. Find the pdf of X. There are 500 students at this uni— versity, find the expected revenue of the library from copying and soda machines. [10 points] 4. A company produces two types of memory chips, its own brand Zeta and a generic brand. The life times (in hours of usage) of each of these brand is a normally dis- tributed random variable with mean ,u and standard deviation 0 given below. For the Brand Zeta , a = 8, 000 hours and a = 400 hours. For the generic brand, ,u = 8, 000 hours and a = 800 miles. It is known that 80% of all chips produced by this company are Brand Zeta, and 20% of all chips produced by this company are generic brand. Let Z denote the event that a randomly selected chip from company’s production is Brand Zeta. Let A denote the event that a chip fails at 8800 hours. (a) Find the probability of the event A for the brand name chip Zeta, i.e., P(A[Z). [8 points] (b) Find the probability of the event A for the generic brand chip, i.e., P(A|ZC). [8 points] (c) Find the probability of the event A, P(A). [6 points] (d) Given that a chip sold by this company failed in 8,800 hours, (i.e, the event A occurred), find the conditional probability that it was a Brand Zeta chip. [8 points] ...
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