Exam 2 for Applied Probability - 2011 | APPM 3570, Exams of Probability and Statistics

Download Exam 2 for Applied Probability - 2011 | APPM 3570 and more Exams Probability and Statistics in PDF only on Docsity! APPM 3570 — Exam #2 — April 6, 2011 On the front of your bluebook, print your name, student number and a grading table. Define all notation, events and random variables that you use. Correct answers with no supporting work may receive little or no credit. You may use a calculator but no other electronic device. At the end of the exam, please sign the honor code pledge printed on your bluebook. 1. (27 points) A few short answer questions to start: (a) Suppose you know E(X) = 3 and V (X) = 2 for some random variable X. Find (i) E(X2), (ii) E(4X + 5), and (iii) V (4X + 5). (b) The probability that Ms. Brown will sell a piece of property at a profit of $3000 is 3/20, the probability that she will sell it at a profit of $1500 is 7/20, the probability that she will break even is 7/20, and the probability that she will lose $1500 is 3/20. What is her expected profit? (c) A Rayleigh random variable, W , has cumulative distribution function F (t) = 1 − e−bt2/2 where t ≥ 0 and b is a fixed constant. (F (t) = 0 if t < 0.) Find the hazard rate function for W . 2. (25 points) A bookshelf contains eight distinct books, three are probability books, two are math books, and 3 are physics books. Two books are chosen at random. Let X be the number of probability books that are chosen. Let Y be the number of math books that are chosen. (a) Find the probability mass function for X. (You can do this part and just ignore Y .) (b) Find the joint probability mass function of X and Y . (Hint: What can you do to verify your answer is reasonable?) (c) Are X and Y independent? (Explain using the definition for two discrete random variables to be independent.) 3. (28 points) When you put your money into a coffee machine, a paper cup comes down and coffee is dispensed into the cup. You are supposed to get 8 oz of coffee. However, the actual amount of coffee dispensed is a random variable with a normal distribution with µ equal to the machine setting and σ = 0.25 oz. (a) What should the machine setting of µ be so that, in the long run, only 2% of the drinks will contain less than 8 oz? (b) Now, suppose you set µ = 8.25 (and σ is still 0.25). What is the probability that a randomly selected cup holds less than 8 oz? (c) If you sell 10,000 cups of coffee in the month, estimate the probability that 2000 or fewer cups will hold less than 8 oz. 4. (20 points) Suppose the joint density of two random variables X and Y is given by f(x, y) = { c(x+ y) for 0 ≤ x ≤ y ≤ 2 0 else. (a) Find the value of c. (b) Find P(Y ≥ 1). Extra Credit (5 points) Find the joint distribution function, FX,Y (x, y) for all values of x and y for the density function given in problem 4. ri N h I T h II \ — N _ _ _ r N _ _ _ C S _ _ I) C C U N 1 I ‘ c S 0 _ _ j V _ . 5 T h N H _ _ _ I _ _ _ \ N 0 _ _ _ _ 5 , I’ I p _ _ _ ‘ I , 1 _ 0 ‘ I — — U C ii II d 4 5 4 I .