STAT 418 MID-TERM EXAMINATION 2 FALL, 2008NAME (in BIG BLOCK letters):SIGNATURE:Student Number:Instructions:1. Please verify that your exam paper contains all seven questions.2. You may use a calculator and one 8.5 × 11 sheet of notes.3. Illegible handwriting will not be graded.4. To earn partial credit, please justify your answers and show all your work.5. Your instructor will answer no ques tions during the examination period. Simply explainthe nature of any error you find in a question and proceed to the next question.DO NOT WRITE BELOW THIS LINEQuestion Marks1234567TotalQuestion 1. [This question is based on Problem 1, p. 187 in the textbook.]Two balls are chosen at random from an urn containing 2 red, 4 white, and 3 green balls.Supp ose that we win $2 for each green ball selected, a nd we get nothing for any red orwhite ball selected. Let X be the amount of money we win.(a) What are the possible values of X?(b) What are the probabil ities associated with each possible value of X?(c) Prove that E(X) = 4/3 and Var(X) = 14/9.(d) Ludo’s Casi no offers this game to the public, charging a fee of $1.01 cents per play. IfCindy plays the game 100 times, what is her expected profit (or loss)?Question 2. [This question is based on Problem 40, p. 192 in the textbook.]After a weekend of Homecoming and Halloween parties, a student named Freddy K. takesa multiple-choice quiz with 5 questions, each question having 3 possible answers. On eachquestion, Freddy chooses an answer at random. Let X be the number o f questions whichFreddy answers correctly. [Yo u may assume that individual questions are independent ofeach other.](a) What is the probability distribution of X? (Please give a complete explanati on foryour answer.)(b) Prove that the probability that Freddy answers at least 4 questions correctly is 11/243.(c) Suppose Freddy gets 2 points for each correct answer and loses 1 point for each incorrectanswer. If Y is Freddy’s total score on the exam then calculate E(Y ). [Hint: Express Yin terms of X, and use that relationship to compute E(Y ) from E(X).]Question 3. [This question is based on Problem 57, p. 177 in the textbook.]Supp ose that the number of a ccidents occurring daily on a highway is a Poisson randomvariabl e with parameter λ = 4.(a) Find the probability that no accidents occur today.(b) Find the probability that 2 or more accidents occur today.(c) Repeat part ( b) under the assumption that at least 1 accident has already occurredtoday.Question 4. [This question is based on Problem 4, p. 255 in the textbook.]A continuous random variable X has probability density functionf(x)=!a + bx4, 0 ≤ x ≤ 10, otherwiseWe are also given that E(X) = 5/6.(a) Find the values of a and b.(b) Evaluat e the standard deviation of X.(c) Calculate P (X<1/2).Question 5. [This question is based on Problem 32, p. 231 in the textbook.]The time T (in hours) needed to repair a car is an exponential random variable withparameter λ =1/2.(a) Prove that P (T>2) = 0.367.(b) Prove that the probability t hat T exceeds 10 hours, given that T has already exceeded9 hours, is 0.606.(c) Prove that T satisfies the lack-of-memory property.Question 6. [This question is based on Self-Test Problem 8, Chapter 5, in the textbook.]Let X be the IQ score of a ra ndoml y chosen IQ test-ta ker. It is known that X is normallydistributed with mean µ = 100 and standard deviation σ = 15. What is the probabilitythat the IQ test score of such a person is:(a) Above 125?(b) Between 90 and 110?(c) Exactly 102?(d) Above 100 given that it is between 9 0 and 11 0?Question 7. [This bonus problem is based on classroom discussions.](a) What are the Four Horsemen?(b) In which year was Zeno of Alea, the Greek philosopher, born?(c) Two people, one of age 21 and the other of age 47 years, are chosen at random. Which ofthe two has a higher probability of surviving to age 48 ? Why? [Hint: If X is the l ife-lengthof a randoml y chosen person, prove that P (X ≥ 48|X ≥ 47) ≥ P (X ≥ 48|X ≥ 21).