1801_0102sem1

1801_0102sem1 - THE UNIVERSITY OF HON G KONG DEPARTMENT OF...

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Unformatted text preview: THE UNIVERSITY OF HON G KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1000 PRINCIPLES OF STATISTICS STAT1801 PROBABILITY AND STATISTICS: FOUNDATIONS 0F ACTUARIAL SCIENCE December 19., 2001 Time: 2:30 p.111. — 4:30 p.111. 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. It is the candidate ’3 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 esamination 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 uiolation of the criteria listed above. Answer ANY FIVE questions. Questions are of equal value. 1. (a) Consider three events AB and C’ with P(A) = 0.5,P(B) = 0.4 and P(C) = 0.5. Suppose that these events are pairwise independent and the probability that an outcome belongs to both A and B but not C is 0.1. (i) Compute the probability that an outcome belongs to all three events. Do you think that these three events are independent? (ii) Compute the probability that an outcome belongs to only two events. (iii) Compute the probability that an outcome belongs to none of the three events. (iv) Given that an outcome belongs to either A or B, or both. compute the probability that it also belongs to C. Hint: A Venn diagram will be useful for the above calculations. (b) A telecommunication company carries out a survey on the reliability of three networks provided to its mobile phone users. It is known that 20% of users subscribe to Network A, 40% to Network B and 40% to Network 0. (i) The average revenues per user per month of Networks A,B, and C are $238, $168 and $98, respectively. Compute the average revenue per user of the telecommunication company in a particular month. sans: STATIOOO Principle of Stat./STAT1301 Probability andIStat. 2 (ii) The customer service manger receives complaints about poor network connections from the users. He finds that 5%, 10% and 15% of users of Networks A, B, and C complain about poor connections. If the manager receives a letter of complaint, compute the probability that this complaint comes from a Network A user. 2. An urn contains a white ball and two black balls. A ball is randomly drawn from the urn with (i) If X denotes the number of draws up to and including the first black ball, write down the probability mass function of X . Hence compute the probability that the black ball appears after the 5th draw. (ii) If Y denotes the of draws up to and including the kth black ball, write down the probability mass function of Y. Prove the probability that the third black ball will turn up at the 5th draw is 16/81. (iii) Given that the probability generating function of X in is 23 deduce the probability generating function of Y in (ii). Use this proba— bility generating function to verify the probability obtained in (ii). (iv) Approximate the probability that the black ball was drawn at least 60 times in 100 independent draws. Hint: You may use the result that DO . k—1+t .. (1 - qs)‘k = E ( k 1 ) qist where q = 1 —- p. i=0 3. (a) Suppose that X; and X2 are two independent discrete random variables, each uniformly distributed on 1, 2,. . . ,n with probability mass function 1 . P(X,-=;r)=E i=1,2,x=1,21_”3n, (i) Find the probability generating functions of X1 and X2. (ii) Deduce the probability generating function of Z = X1 +X2 and hence show that the mean of Z is rt + 1. (b) A box contains coloured marbles of which the proportion of red mar- bles is p. w. marbles were drawn with replacement from the box and X red marbles were observed. Obviously, X follows a binomial Births, p) distribution. S&AS: STATIOOO Principle of Stat. / STAT1801 Probability and Stat. 3 (i) Show that the moment generating function of X is given by M(t) 2* (pet + q)” where q = 1 —p. Hence, obtain the mean and variance of X. (ii) State the conditions under which the binomial probability distribu- tion can be approximated by the Poisson distribution. If n = 500 and p = 0.01, compute the probability P(3 < X <2 7). 4. The lifetime (in hours) of an electronic component is assumed to follow an exponential distribution with probability density function f(33)=-)1:e"$/)‘ $>0,A>0. where A is unknown. (1) Compute the probability that the component is still working after 2A hours. (ii) Obtain the mean and the variance of X. (iii) A random sample of to components gives lifetimes X 1, X 2, . . . , X”. Obtain a point estimator of A, say A. (iv) State the Central Limit Theorem and use it to compute the approximate probability that the mean lifetime is longer than 5.75 hours when n = 100 and A = 5. You may use the result in (iii). 5. (a) LBlJ X1, X2, . . . 1X“ distribution with unknown )1 and or. (1) Show that the estimator X = $2}; Xi is unbiased for a. Write down the sampling distribution of X. size n from a normal N (p.02) (ii) Write down the minimum variance unbiased estimator for o. (b) The branch manager of a local bank would like to study the queueing times of his customers. He randomly selected 15 customers and their queueing times were recorded. The observed sample mean and sample standard deviation of the queueing times were 8.0 minutes and 3.0 min- utes, respectively. To improve the service to his customers. he provided an intensive training course to his staff. After the training course, he observed queueing times of 27 randomly selected customers. The sample mean and sample standard deviation were 6.0 minutes and 2.0 minutes, respectively. You may assume that the queueing times before and after the training course follow normal distributions with means (11 and M2 and a common variance 02. (i) Construct a 95% confidence interval for M ~— #2. SECAS: STATIDOO Principle of Stat. / STAT1801 Probability and Stat. 4 (ii) Test, at the 1% level of significance, whether it is true to say that the service provided by the bank has been improved. Note: In testing hypothesis, state clearly the hypotheses of interest, the distri- bution of the test statistic, the decision rule and your conclusion. 6. A survey is carried out to study the proportion of students at a local university who have experience in Internet shopping. Of 100 students interviewed, 40 had shopped on the Internet. (i) Construct a 95% confidence interval for the true proportion of students who have shopped on the Internet. Is this interval exact? Explain. (ii) Is there any evidence that more than 1/3 of the local university students Shep on the Internet? Draw your conclusion based on the 5% level of significance. (iii) A similar survey was carried out at an overseas university and 260 stu- dents out of 500 had shopped on the Internet. Can you draw a conclusion that Internet shopping is more popular for overseas university students than for local university students? Draw your conclusion based on the 5% level of significance. Note: In testing hypothesis, state clearly the hypotheses of interest, the distrin bution of the test statistic, the decision rule and your conclusion. 88.5.48: STAT1000 Principle of Stat./STAT1801 Probability and Stat. Table Standard Narmal Curve Areas (tum u m: s z} z 0.00 0.0! 0.02 0.03 0.04 0.05 0.06 0.07 0. 03 0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.633 1 0.6368 0.6406 0.6443 0.6480 0.65 1 7 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.73 89 0.7422 0.7454 0.7486 0.75 17 0.7549 0.7 0.7580 0.761 1‘ 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133. 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 . 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1 .3 0.9032 0.9049 0.9066 0.9082 0.9099 0.91 15 0.9 131 0.9 147 0.9 162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9278 0.9292 0.9306 0.93 19 1 .5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.941 8 0.9429 0.9441 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.95 15 0.9525 ‘ 0.9535 0.9545 1.7 0.9554 0.9564 0.9573 0.95 82 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.964 1 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 . 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.98 17 2. 1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 _ 0.9890 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.991 1 0.9913 0.9916 2.4 0.99 1 8 0.9920 0.9922 0.9925 0.9927 0.9929 0.993 1 0.9932 0.9934 0.9936 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 - 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 ‘ 0.9974 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.998 1 2.9 0.998 1 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 3. 1 0.9990 0.9991 0.9991 0.999 1 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 . 0.9995 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 3-4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 SfizAS: STATIOOO Principle of Stat. / STATlBOl Probability and Stat. Table Critical Valuu 1.“. far tin t Damnation a v .10 .05 .025 .01 .005 .00! .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 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 I 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 1325 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 H 1.282 1.645 1.960 2.326 2.576 3.090 3.291 m 6 ...
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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_0102sem1 - THE UNIVERSITY OF HON G KONG DEPARTMENT OF...

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