1801_0405sem1

1801_0405sem1 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1801 PROBABILITY AND STATISTICS: FOUNDATIONS OF ACTUARIAL SCIENCE December 14, 2004 Time: 2:30 p.m. — 4:30 p.m. Candidates taking examinations that permit the use of calculators may use any calculator which fulfils the following criteria: (a) it should be self-contained, silent, battery-operated and pocket—sized and (b) it should have numeral-display facilities only and should be used only for the purposes of calculation. It is the candidate ’s responsibility to ensure that the calculator operates satisfacto- rily and the candidate must record the name and type of the calculator on the front page of the examination scripts. Lists of permitted/prohibited calculators will not be made available to candidates for reference, and the onus will be on the candi- date to ensure that the calculator used will not be in violation of the criteria listed above. Answer ALL questions. Marks are shown in square brackets. 1. (a) An auto insurance company classifies its policyholders as good, average, or bad risks: 30% are deemed good risks, 50% are deemed average risks, and 20% are deemed bad risks. Historical data suggest that 5% of the good risks, 10% of the average risks, and 40% of the bad risks will be involved in an accident in the coming year. (i) What is the probability that a randomly chosen customer files an acci- dent claim in the coming year? (ii) An accident claim has just been filed with the company. What is the probability that this customer was classified as a bad risk? (iii) The company would like to have accident claims on at most 10% of its policies. Consequently, it decides to cancel the policies of some bad risk customers and replace these policies with average risks. Of the company’s customers, 30% will remain classified as good risks. What is the smallest percentage of the company’s customers who must be classified as average risks for the proportion of customers filing accident Claims to be not more than 10%? S&AS: STAT1801 Prob. and Stat.: Foundations of Actuarial Science (b) A car-hire service company knows that on any given day, the number of persons, X, who want to hire a car from it follows the Poisson probability distribution: ~5.5:c P(a:)=e , where x=0,1,2,.... :13! The daily all—inclusive cost of keeping a car is $350, whether it is hired or not. If the car is hired, the company collects a rental of $700 per day. The company is only willing to keep not fewer than 4 but not more than 6 cars. (i) Calculate the company’s expected daily profit for each of the cases of keeping 4, 5 and 6 cars. [Notez P(O) = 0.0067, P(l) = 0.0337, P(2) = 0.0842, P(3) = 0.1404, P(4) = 0.1755, P(5) = 0.1755, P(6 or above) 2 0.3840] (ii) Hence determine the optimum number of cars that the company should keep in order that the expected daily profit is maximized. [Total: 17 marks] 2. An insurance company wants to study if its clients are satisfied with the service when they claim their medical insurance. The clients are asked to score their satisfaction level in the range of 0~100 (where 0 means totally unsatisfied, and 100 means totally satisfied). The company wants to see if its service is better than the industry-wide service which has an average score of 75. The company of course cannot claim a better service unless there is sufficient evidence to say so. Suppose that the company collects a sample of n = 16 clients, and the standard deviation of the scores is 0 == 18. (a) State the null and alterative hypotheses. (b) Give the decision rule for testing the hypotheses at the 5% significance level. (c) If the true population mean score for the clients of the company is 85, what is the power of the test you have constructed in (b)? (d) The scores obtained are listed as follows: 65, 100, 100, 51, 87, 67, 100, 57, 90, 87, 100, 100, 95, 100, 48, 89. What is your conclusion on the hypotheses? (e) State the assumptions you have used. Are the assumptions reasonable or not? [Total: 16 marks] S&AS: STAT1801 Prob. and Stat.: Foundations of Actuarial Science 3. (a) Suppose that X and Y are random variables with joint density 6(1—33—y) forx+y£1,x20,y20 flay): elsewhere Determine the distribution of the sum Z = X + Y. (b) Suppose that X and Y are random variables with joint density 1 forOSxSl,OSy§l flay) ={ 0 elsewhere Use the Jacobian transformation technique or otherwise to determine the distribution of the sum Z = X + Y. [Total: 22 marks] 4. A researcher used two independent groups of band B students who were known to excel in mathematics. He assigned 11 students to a control group that was simply asked to complete a difficult mathematics examination. He also assigned 12 students to a threat condition (experimental group), in which they were told that band A students typically did better than other students in mathematics tests, and that the purpose of the examination was to help the experimenter to understand why this difference exists. The researcher reasoned that simply telling band B students that band A students did better on mathematics tests would arouse feelings of “stereotype threat” and diminish the students’ performance. The dependent variable is the number of items correctly answered. The experimental data are shown below. Control subjects: 4, 9, l2, 8, 9, 13, 12, 13, 13, 7, 6. Mean :31 = 9.64; st. dev. 51 = 3.17; m :11. Threat subjects: 7, 8, 7, 2, 6, 9, 7, 10, 5, 0, 10, 8. Mean 552 = 6.58; st. dev. .32 = 3.03; 72.2 = 12. (a) State the null and alternative hypotheses. Construct a test for testing the hypotheses at the 5% level of significance. (b) Using plain language and in no more than 100 words, write up the results of the experiment. (c) State the assumptions you have used in (a). Are they reasonable or not? (01) Let p D be the population mean difference of the number of items correctly solved by the control subjects and that by the threat subjects. If one wants 3 S&AS: STAT1801 Prob. and Stat.: Foundations of Actuarial Science the width of the 95% confidence interval for ,up to be less than 2, what is the minimum sample size n in each group (n is the same in both groups) that he should take? You may use the pooled variance estimate as if it were the true population variance. [Total: 18 marks] 5. The length of life X (in hours) for a type of electric component has an expo- nential distribution with a density function given by f() x20 3:: 0 elsewhere (a) Derive the moment generating function for X. Hence determine its mean and variance. (b) Evaluate P(1 g X < 2). (c) It is known that a component has survived 1 hour already, what is the probability that it will survive two hours? (d) Let X1 and X2 be the lifetimes for two components. Find the moment generating function for Y = X = (X 1 + X2) / 2. What is the probability distribution of Y? Give its density function. (e) Give an exact result for P(1 g Y < 2). (f) Also give an approximate result for P(1 S Y 3 2) assuming the central limit theorem. Is it a good approximation? Give your reason(s). [Total: 27 marks] ...
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This note was uploaded on 03/18/2012 for the course STAT 1801 taught by Professor Mrchung during the Fall '10 term at HKU.

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1801_0405sem1 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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