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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 selfcontained, silent,
batteryoperated and pocketsized 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
34 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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 Statistics, Probability

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