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Faculty Of Computer Studies M130 Introduction to Probability and Statistics FINAL EXAMINATION Fall / 2013 – 2014 Number of Exam Pages (including this cover sheet): 4 Time Allowed: 2 Hours Keys Instructions: Please read the following instructions before starting: 1. This is a closed book exam. 2. This exam accounts to 50% of the total mark in the course. 3. Budget your time according to the mark assigned to each question. 4. Mobile phones and all other mobile communication equipments are NOT allowed. 1 Part 1: MULTIPLE CHOICE QUESTIONS You may solve all questions. Each question is worth 2 marks. Your grade in this part is that of the best 5 questions, giving a total of 10 possible marks. Q–1: A box contains 9 tickets numbered 1,2,…,9. If one ticket is drawn at random, the probability that the number on the ticket is more than 3 is: a) 1/2 b) 1/3 c) 2/3 d) 9/3 e) None of the above Q–2: How many permutations are there of the letters of the word STATISTICS? 10! 3!.3! 10! b) 3!.3!.2! a) c) 10! d) 10! 3!.2! e) None of the above Q–3: If a pair of fair dice is rolled, then the probability of getting a total of 6 is a) 1/6 b) 7/36 c) 5/36 d) 12/36 e) None of the above ax, 0 x 1 is a density function, then a = 0, otherwise Q–4: If f x a) b) c) d) e) 2 0 1 -2 None of the above Q–5: Which of the following can’t be the probability of an event? 2 a) b) c) d) e) -4/7 0.0069 1 0 None of the above Q–6: The following is known to be a probability distribution: x 0 1 2 3 p(X=x) ? 0.25 0.35 0.20 The probability that x=0 is: a) 0 b) 0.5 c) 1 d) 0.05 e) None of the above 4 0.15 Part 2: ESSAY QUESTIONS Each question is worth 10 marks. You should answer only FOUR out of the five questions, giving a total of 40 possible marks. Q–1: [5+5 Marks] One bag contains 5 red balls and 4 black balls, and a second bag contains 4 red balls and 6 black balls. One ball is drawn from the first bag and placed unseen in the second bag. a) What is the probability that a ball drawn from the second bag is black? p B2 4 7 5 6 58 . . 9 11 9 11 99 b) What is the probability that the first ball is black given that a ball drawn from the second bag is black? 4 7 . P( B2 B1 ) 9 11 28 PB1 / B2 58 P B2 58 99 Q–2: [2+2+2+4 Marks] Consider the experiment of tossing a coin 3 times. Let the variable x be the number of tails. 3 a) Find the probability distribution for the number of tails. x 0 1 2 3 p(X=x)=f(x) 1/8 3/8 3/8 1/8 b) Compute p(x 2) 2 P X 2 f x 1 3 3 7 8 8 8 8 x 0 c) Find the expected value for the number of tails. 3 1 3 3 1 3 E X xf x 0 1 2 3 8 8 8 8 2 x 0 d) Find the variance and the standard deviation for the number of tails. 2 E ( x 2 ) ( E ( x)) 2 3 3 1 24 E ( x 2 ) 0 1 4 9 3 8 8 8 8 9 3 2 3 4 4 3 0.866 4 Q–3: [3+2+2+3 Marks] Suppose x is a random variable with density function given by: 1 x 0 x2 f x 2 0 otherwise a) Find the cumulative distribution function. x x x t 2 1 x2 F ( x) F (t )dt t dt 2 4 0 4 0 if x 0 0 2 x Hence, F ( x) if 0 x 2 4 1 if x 2 b) Find p(x<1) and p(1<x<1.1) 4 p( x 1) F (1) 1 4 2 1.1 p(1 x 1.1) F (1.1) F (1) 4 1 0.0525 4 c) Find the mean of the random variable X. 2 x3 4 1 E X x x dx 2 6 0 3 0 2 d) Find the variance of the random variable X. E X2 2 x4 2 1 x x dx 2 2 8 0 0 2 2 E X 2 2 2 16 2 9 9 Q–4: [3+5+2 Marks] a) Suppose x is a binomial random variable with n = 4 and p=1/3. Find p(x=3) 3 1 1 2 4! 1 2 8 p 3;4, 4C3 . . 3 3 3 3!1! 27 3 81 b) A fair coin is tossed 8 times. Find the probability of getting at least 2 heads. p( x 2) 1 p( x 2) 1 p( x 0) p( x 1) 8 1 0 1 8 8 1 1 7 1 0 2 2 1 2 2 9 247 1 256 256 c) Find the coefficient of x 5 in the expression of 2 x 58 . 224,000 Q–5: [3+3+4 Marks] The weekly salaries of 5,000 employees of a large corporation are assumed to be normally distributed with mean $450 and standard deviation $40. Hint: use the below table. 5 Z 0.7 0.75 0.8 1 1.29 1.35 2 p(Z<z) 0.7580 0.7734 0.7881 0.8413 0.9015 0.9115 0.9772 a) If an employee is selected at random, find the probability that he or she makes less than $480 . P X 480 ? x 480 450 3 z 0.75 40 4 p( x 480) p ( z 0.75) 0.7734 b) Find the probability that he or she makes between $480 and $530. 530 450 z1 0.75 z2 2 40 p (480 x 530) p (0.75 z 2) p ( z 2) p ( z 0.75) 0.9772 0.7734 0.2038 c) Find the salary below which exist 90% of the employees’ salaries. P( Z z ) 0.9 from the table z 1.29 Z x x Z 40(1.29 450) $501.6 6

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