Download Statistics 400 Final Examination and more Exams Probability and Statistics in PDF only on Docsity! Statistics 400 (Dr. Rosenberg) Final Examination December 14, 1996 Instructions. Answer all questions. The point value of each problem is indicated. The exam is worth a total of 100 points. Calculators and textbooks are allowed, but not laptop computers or notes. A few problems may require use of the tables at the back of the textbook. In problems with multiple parts, the parts are graded independently of one another. Be sure to go on to subsequent parts even if there is some part you cannot do. You may assume the answer to any part in subsequent parts of the same problem. Please be sure to write down intermediate steps of calculations, not just the final answer. An unjustified correct answer may not receive full credit. Also, the more work you show, the better your chances of getting partial credit if your final answer to a problem is not completely correct. You may leave answers in terms of exponentials, logarithms, etc. 1. (15 points) A poker hand of 5 cards is drawn at random from a standard deck of 52 cards (consisting of 4 suits, each containing cards numbered 2 through 10, J, Q, K, and A). (a) What is the probability that the hand contains all four aces? (You may write the answer in terms of factorials or binomial coefficients.) (b) What is the probability that the hand is "four of a kind," i.e., contains all four aces, or all four kings, or all four queens, ..., or all four 2's? (You may write the answer in terms of factorials or binomial coefficients.) 2. (15 points) A standard (fair) die (with faces numbered 1 through 6) is rolled 100 times, and the results of the rolls are added. Use the Central Limit Theorem (and the continuity correction) to approximate the probability that the sum is precisely 350. 3. (15 points) Suppose X and Y are independent normal random variables with E(X) = 2, E(Y) = 4, V(X) = 16, and V(Y) = 9. (a) Compute the mean and variance of X —Y. (b) Compute P(X > Y). (Use your answer to (a).) (c) Compute P(X > 3 and Y < 3). 4. (15 points) Suppose Xi, ... , Xn are independent, identically distributed random variables, each with probability density function f(x) = (a — l)x~ Q , 1 < x < oo, where a > 1 is a fixed parameter. (a) (10 points) If .X i, ... , Xn are observed to take the values xi, ... , a: n , respec- tively, find the likelihood function for a in terms of #1, ... , x n . Then find the maximum likelihood estimator for a. in terms of X±, ... , Xn . (Hint: take the natural logarithm of the likelihood function before differentiating.) 1
