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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 selfcontained, silent,
batteryoperated and pocket—sized and (b) it should have numeraldisplay 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 carhire 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 industrywide 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.
 Fall '10
 MrChung
 Statistics, Probability

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