Engineering Statistics and Data Analysis - Exam 1 with Answers | IMSE 321, Exams of Systems Engineering

Download Engineering Statistics and Data Analysis - Exam 1 with Answers | IMSE 321 and more Exams Systems Engineering in PDF only on Docsity! Exam –I, Fall 2005 1/6 Engineering Statistics and Data Analysis (IMSE 321) Name:-------------------------------------- IMSE/UNL Fall 2005 (Sep. 19/2005) Exam I 40 minutes, Points + Bonus, Solve 2 problems out of last three (4,5, and 6) 1. In a game, a player tosses a die and then flips a coin. The player wins if the outcomes are odd and head, respectively, and loses if the outcomes are even and tail. Let the pair of random variables (X, Y) represent the outcomes of the die and coin (X for the die and Y for the coin). Also let S, W, and L respectively denote the outcome space, the event in which player wins, and the event in which player loses. (a) Which of the following statements is NOT correct? i. S={(1,H), (2,H), (3,H), (4,H), (5,H), (6,H), (1,T), (2,T), (3,T), (4,T), (5,T), (6,T)} ii. S={(1,0), (2,0), (3,0), (4,0), (5,0), (6,0), (1,1), (2,1), (3,1), (4,1), (5,1), (6,1)} where Y=0 denotes head and Y=1 denotes tail. iii. S={(x,y): x=1,2,3,4,5,6, and y=0,1} where Y=0 denotes head and Y=1 denotes tail. iv. S={(1,0), (2,0), (3,0), (4,0), (5,0), (6,0), (1,1), (2,1), (3,1), (4,1), (5,1), (6,1)} where Y=1 denotes head and Y=0 denotes tail. (b) Let T denote the event that game is a tie (The player neither wins nor loses). Which one of the following statements is correct? i. T=∅ ii. T=W′∪L′ iii. T=(W∪L)′ iv. T=S∩(W′∪L′) (c) Let M be the event in which die turns less then four; i.e. M={(x,y): (x,y)∈S, x<4}. Which statement does NOT represent the event W|M? i. The player won the game given the die turned less than four. ii. The die turned less than four, also the player won. iii. The die did not turn more than three, also the player won. iv. The die turned less than four and the player won. (d) P(W|L) is equal to i. 3/3. ii. 3/6. iii. 3/12. iv. None of the above. (e) P(W|M) is equal to i. 3/3. ii. 3/6. iii. 2/6. iv. 2/12. 10 100 25 5 5 5 5 5 Exam –I, Fall 2005 2/6 2. X and Y are random variables with means µx and µy, variances σx 2 and σy 2, and a covariance σxy. For the following functions of X and Y give the appropriate mean and variance using the terms µx, µy, σx 2, σy 2, and σxy. (a) X/3 (b) Y-X (if X and Y are not independent) (c) 2Y+3X (if X and Y are not independent) (d) Y-X (if X and Y are independent) (e) 2Y+3X (if X and Y are independent) 25 5 5 5 5 5 Exam –I, Fall 2005 5/6 5. A process has the following joint probability density function of the random variables X and Y: f(x,y)=2/3(x+2y) for 0< x <1 and 0< y <1. (a) Find P(0<X<.5, 0.25<Y). (Set up completely, do not integrate) (b) Find the marginal distributions of X and Y. (c) Are X and Y independent? Why? (d) Find the expected value of X - 4Y. (Set up completely, do not integrate) 15 5 5 5 5 5 Exam –I, Fall 2005 6/6 6. There are five chairs in a row and three persons are going to sit in that row. (a) Find the number of different combinations that three chairs can be occupied (The order of persons is not important). (b) Find the number of different ways that three person can be seated in that row when the order is important. (c) What is the probability of the case that no two persons are sitting by each other? (d) If you observe a sitting configuration similar to part c, would you conclude that those people have selected their chairs randomly? Explain. 15 5 5 5 5 5