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Uppsala University Department for Information Science Statistics B1 2013-03-16 WRITTEN EXAM FOR THE COURSE, PROBABILITY THEORY AND STATISTICAL INFERENCE, B1 (7.5 ECTS) Writing time: 0900-1300 Permitted aids: Formulas for the course Probability Theory and Statistical Inference Math-Handout, Lars Forsberg Pocket calculator Dictionary (or word-list) Notations in the permitted aids are not allowed. The written examination has 5 problems, for a total of 100 points. If you desire clarification regarding the test, especially the wording of a problem, then please alert an examination proctor. The examination proctors can contact the responsible instructor. After turning in your test, you may keep the test-pages with the question-statements. INSTRUCTIONS: A. Carefully follow the instructions that are listed on the examination-directions page. B. State the assumptions that must be made for the method to be applicable. C. Account for every essential step in your solution. If special concerns are raised in the problem statement, then your solution must carefully address those concerns. 21/13 Task 1. (20p) A small firm with 20 employees manufactures insulation. The firm is requested to send three employees who have positive indications of asbestos on to a medical center for further testing. Therefore a number of employees need to be tested for positive indications of asbestos. For that reason a simple random sample (without replacement) is taken. Assume that 8 of the 20 employees have positive indications of asbestos in their lungs and calculate the following probabilities. a) If the sample size is ten what is the probability that the sample will contain at least three employees with positive indications of asbestos. b) Find the probability that ten employees must be tested in order to find three positives, that means the probability to get the third positive at the 10th drawing. Task 2. (20p) a) Assume that X1, X2, and X3 are independent random variables with zero mean and equal variances = V(X). Let Y1 = X1 + X2 - 3X3 and Y2 = X1 - X2 +3X3. Find the variance of Y1 and Y2. b) Find the correlation between Y1 and Y2. c) Let Y be chi-square distributed with 10 degrees of freedom. Find the expected value of X = 60 – 10Y + Y2. Task 3. (20p) Given: k ( y y2 ), f ( y1 , y2 ) = 1 0 0 y1 1, 0 y2 1 elsewhere. a) Demonstrate that k need to be equal to one in order for f(y1,y2) to be a density function. b) Find the marginal density function for Y1 and E(Y1). c) Find the median for Y1 (See footnote 1). Task 4. (20p) Let Y1 ,Y2 ,Yn denote a random sample from the Poisson distribution with mean λ. a) Find the MLE estimator for λ. b) What is the MLE for P(Y ≤1)? Task 5. (20p) Given the following results obtained from two independent samples from normally distributed populations with means µy and µx: ny = 8, nx = 11, y 9.3 and x 8.1 , s2y =2.2 and sx2 unknown. a) Test H 0 : y x against H1 : y x under the assumption that s x2 is unknown. Present hypothesis, test statistic, assumptions, rejection region and conclusion in detail. b) An important assumption for the test in Task a) is that the two population variances are equal. Test the assumption of equal population variances given that s x2 =1.5. 1) 𝑝 𝑝 2 2 2 The solutions to the equation x2+px+q= 0 is given by 𝑥 = − ± √( ) − 𝑞