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# 1801_0001sem1 - THE UNIVERSITY OF HONG KONG DEPARTMENT or...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT or STATISTICS AND ACTUARIAL SCIENCE STAT1000 PRINCIPLES OF STATISTICS STAT1801 PROBABILITY AND STATISTICS: FOUNDATIONS OF ACTUARIAL SCIENCE December 20, 2000 Time: 9:30 a.m. - 11:30 a.rn. Candidates taking examination that permit the use of calculators may use any cal- culator"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 purpose of calculation. I t is the candidate’s responsibility to ensure that the calculator operates satisfactorily 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 candidate to ensure that the calculator used will not be in violation of the criteria listed above. Answer ANY FIVE questions. Questions are of equal value. 1. A certain machine consists of 3 components, A, B, and C. In 10,000 routine examinations of this machine defects in A occurred 500 times, defects in B 400 times, defects in C 200 times, while defects in both A and B but not C occurred 15 times. Suppose that the defects of components are pairwise independent. Let P(A), P(«B) and P(C') be the probabilities of defects in A, B and C, respectively. (i) Write down the values for P(A), P(B), and P(C'). (ii) Compute the probability that the machine has defects in all three compo— nents in a routine examination. Are the defects of the three components independent? (iii) Compute the probability that machine has defects in only two compo- nents in a routine examination. (iv) What is the probability that the machine has no defect in a routine examination? (v) If it is known that the machine has defects in either component A or component B, or both, what is the probability that it has also a defect in component C? \ . . . t , . m-............._ m.“ ....._.y....m--mtm«.._ , .._.,.__.__4...m_....u...,.. .r ...W....t..w.m...w .. S&AS: STATlOOO Principle of Stat./STAT1801 Probability and Stat. 2 (vi) Suppose that the replacement costs of components A, B, and C are \$300, \$500 and \$1,000, respectively. What is the expected replacement cost of components in a routine examination of the machine? Hint: A Venn diagram will be useful for the above calculations. 2. Three large urns AB and C each contains a large number of coloured balls of which some are red in colour. The proportions of red balls in urns A, B and C are 1/2, 1/4 and 1/8, respectively. (i) Balls are drawn with replacement from urn A and their colours are recorded. If X denotes the number of draws up to and including the ﬁrst " red ball, write down the probability mass function of X. Hence compute the probability that the ﬁrst red ball appears after the 5th draw. (ii) Balls are drawn with replacement from urn B and their colours are recorded. If X denotes the number of draws up to and including the kth red ball, write down the probability mass function of X. Hence com- pute the probability that the ﬁfth red ball will turn up at the ﬁfteenth draw. (iii) Balls are drawn with replacement from urn C and their colours are recorded. Using the Poisson approximation to approximate the probabil- ity of observing more than ten red balls but less than ﬁfteen red balls in 100 independent draws. State clearly the conditions of using the Poisson approximation. Is the approximation good in this case? (iv) An urn is selected at random and from which a ball is drawn. If the ball is red, compute the probability that it comes from urn A. 3. (a) Consider a discrete random variable X having probability generating function (1 6) S — 3 0X“) ‘ m Obtain the probabilty mass function of X. Give an example which has this probability mass function. (b) Let X be a binomial random variable having a probability mass func- tion [5] < l. P(X = x) = nC';z)“(1—p)"‘z, z = 0,1,. . . ,n. Show that the mean and variance of X are rip and np(1 — p), respec- tively. K- m. .....,. . ,._ . M , . ., .«m-..-m._..cm~.m-um. . . . .wmnmmm... n.........._......~.._..._.,... S&AS: STATIOOO Principle of Stat./STA’I‘1801 Probability and Stat. 3 . (ii) A bag contains 25 marbles of which 10 are blue in colour. A marble is randomly drawn from the bag with replacement. Using the normal approximation, compute the probability that no more than 420 blue marbles are observed in 1000 random draws from the bag. Do you think that this probabilty is close to its true value? Explain. 4. (a) Consider a random variable X having a probability density function f(x) = Ice-“(Ml a: 2 5, A > 0. (i) Show that k = 2A. (ii) Obtain the moment generating function of X. Hence ﬁnd the mean and variance of the distribution. (iii) Find the probability P(X 2 15|X 2 10). Comment on this value. (b) If X is a standard normal random variable, show that its moment gener- p ating function is given by Mx(t) = exp [—319] Furthermore, if X1,X2, . . . ,Xn is a random sample of size n from the standard normal distribution, deduce the moment generating function of the sample mean, X 9- };(Xl + X2 + . .. + X“). Hence write down the distribution of X. 5. A survey is carried out to study the number of hours, X, per day spent on using the Internet by a university student. Suppose that X follows a normal distribution with mean p. hours and standard deviation 0 hours. (a) If it is known that 12.3% of students spend less than 2 hours and 33% of students spend more than 4 hours on the Internet, ﬁnd the values of ,u and a. (b) Suppose that there is an initial charge of \$2.00 for each connection and a usage charge of \$2.50 per hour. Using the results in (a), compute the probability that a student spends more than \$15 on the Internet on a particular day. (c) Sixteen students were randomly selected from University ABC and the numbers of hours per day spent on using the Internet are as follows:- (i) Obtain unbiased estimates of ,u and 02. (ii) Construct a 95% conﬁdence interval for the number of hours per day spent on using the Internet by a student of University ABC. ,.V.._....,...m,....~.. w..-“ W............—.——... ~. . . rm~M_——~m..v._m « . mam—“stun..-” ,m S&AS: STATlOOO Principle of Stat./STAT1801 Probability and Stat. 4 (d) To compare the number of hours per day spent on using the Internet with ~ University ABC, University XYZ took a random sample of 16 students with sample mean and sample standard deviation 5.00 hours and 2.00 hours, respectively. Using the data in (iii) and assuming that the standard deviations of the two universities are equal, (i) show that a pooled estimate of the common variance is 2.9194; (ii) test, at the 5% level of signiﬁcance, whether students from Univer- sity XYZ spend more time on using the Internet than students from University ABC. State clearly the test statistic and the decision rule. 6. Happy Shoppers department store accepts payments by cash or credit cards. On‘a particular day, 479 of the ﬁrst 1000 transactions were made by credit cards. - (a) (b) Construct a 95% conﬁdence interval for the true proportion of transac- tions that were made by credit cards. Is this interval exact? Explain. The manager of the department store would like to study the popularity of using credit cards as a means of payment. He suspects that less than 50% of the transactions are made by credit cards. (i) Write down the null and alternative hypotheses for the manager. (ii) Perform a formal test and draw a conclusion based on the 5% level of signiﬁcance. State clearly the test statistic and the decision rule. The department store launched a campaign to promote the use of credit cards for payment. After the promotion period, 226 transactions out of 400 randomly selected transactions were made by credit cards. Is the campaign successful in promoting the use of credit cards for payment? Draw your conclusion based on the 5% level of signiﬁcance. ********** ********** S&AS: STATIOOO Principle of Stat. /STAT1801 Probability and Stat. 5 Table Standard Normal Curve Areas <l>(z) = Hz 5 z) [Standard normal density function Shaded area = My.) 0.02 0.03 . 0.05 . 0.07 0.08 ‘ “4-....-” QM-“ v- Hm” .mn.,......————W a.“ “www-m—w- .m.’ i. S&AS: STATIOOO Principle of Stat. /STAT1801 Probability and Stat. 6 Table Critical Values ta, for the 1 Distribution 1: v .10 .05 .025 .01 .005 .001 .0005 1 3.078 6.314 12.706 31.821 63.657 318.31 636.62 2 1.886 2.920 4.303 6.965 9.925 - 22.326 31.598 3 1.638 2.353 3.182 4.541 5.841 10.213 12.924 4 1.533 2.132 2.776 3.747 4.604 7.173 8.610 5 1.476 2.015 2.571 3.365 4.032 5.893 6.869 6 1.440 1.943 2.447 3.143 3.707 5.208 5.959 7 1.415 1.895 2.365 2.998 3.499 4.785 5.408 8 1.397 1.860 2.306 2.896 3.355 V 4.501 5.041 9 1.383 1.833 2.262 2.821- 3.250 4.297 4.781 10 1.372 1.812 2.228 2.764 3.169 ' 4.144 4.587 11 1.363 1.796 2.201 2.718 3.106 4.025 4.437 12 1.356 1.782 2.179 2.681 3.055 3.930 4.318 13 1.350 1.771 2.160 2.650 3.012 3.852 4.221 14 1.345 1.761 2.145 2.624 2.977 3.787 4.140 15 1.341 1.753 2.131 2.602 2.947 3.733 4.073 16 1.337 1.746 2.120 2.583 2.921 3.686 4.015 17 1.333 1.740 2.110 2.567 2.898 3.646 3.965 18 1.330 1.734 2.101 2.552 2.878 3.610 3.922 19 1.328 1.729 2.093 2.539 2.861 3.579 3.883 20 1.325 1.725 2.086 2.528 2.845 3.552 3.850 21 1.323 1.721 2.080 2.518 2.831 3.527 3.819 22 1.321 1.717 2.074 2.508 2.819 3.505 3.792 23 1.319 1.714 2.069 2.500 2.807 3.485 3.767 24 1.318 1.711 2.064 2.492 2.797 3.467 3.745 25 1.316 1.708 2.060 2.485 2.787 3.450 3.725 26 1.315 1.706 2.056 2.479 2.779 3.435 3.707 27 1.314 1.703 2.052 2.473 2.771 3.421 3.690 28 1.313 1.701 2.048 2.467 2.763 3.408 3.674 29 1.311 1.699 2.045 2.462 2.756 3.396 3.659 30 1.310 1.697 2.042 2.457 2.750 3.385 3.646 40 1.303 1.684 2.021 2.423 2.704 3.307 3.551 60 1.296 1.671 2.000 2.390 2.660 3.232 3.460 120 1.289 1.658 1.980 2.358 2.617 3.160 3.373 1» 1.282 1.645 1.960 2.326 2.576 3.090 3.291 ...
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