辅导案例-MATH2801

  • August 15, 2020

THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS Semester 1 2016 MATH2801 Theory of Statistics TIME ALLOWED -Two (2) hours TOTAL NUMBER OF QUESTIONS – 4 ANSWER ALL QUESTIONS THE QUESTIONS ARE OF EQUAL VALUE ANSWER EACH QUESTION IN A SEPARATE BOOK THIS PAPER MAY BE RETAINED BY THE CANDIDATE ONLY CALCULATORS WITH AN AFFIXED “UNSW APPROVED” STICKER MAY BE USED Statistical tables are provided at the end of the examination paper. All answers must be written in ink. Except where they are expressly required pencils may only be used for drawing, sketching or graphical work. Semester 1 2016 MATH2801 Page 2 1. [20 marks] Answer this question in a separate book a) Discrete random variables X and Y have the following joint distribution: fx,Y(x, y) y 0 0 4/16 X l 4/16 1 4/16 3/16 2 1/16 0 i) Are X and Y independent? Give a reason for your answer. ii) Consider the product XY. A) Calculate the probability P(XY > 0). B) Compute E(XY) and Var(XY). b) Let X and Y be continuous random variables with joint density: 3 fx,y(x,y) = 2(x 2 + y2 ) for 0 < x < l, 0 < y < l. i) Sketch clearly the region of positive density. ii) Compute the marginal density function fy(y). iii) A) Compute the conditional density fx1Y=ixly). B) Hence determine E(XIY = y). C) Hence determine E(XIY = 0.6). iv) Are X and Y independent? Give a reason for your answer. Please see over ... Semester 1 2016 MATH2801 Page 3 2. [20 marks] Answer this question in a separate book a) An insurance company insures an equal number of male and female clients. In any given year the probability that a randomly chosen male client makes a claim is p1 , and the probability that a randomly chosen female client makes a claim is p2 . Each year a client is chosen at random for a policy review, and the client chosen in any year is independent of the choice in any other year. i) In any year, what is the probability that the randomly chosen client made a claim on their policy in the previous year. ii) What is the probability that, in two successive years, each of the randomly chosen clients made a claim in the year prior to the time in which he/she is selected. iii) Suppose that for two consecutive years each of the randomly chosen clients had made a claim in the year prior to the time in which he/she is selected. What is the conditional probability that both of the clients are female? b) Victor and Wilfred arrive at a railway station independently of one an- other at a random time between 1pm and 2pm. Let V be the number of minutes past 1pm when Victor arrives, and let W be the number of minutes past 1pm when Wilfred arrives. The marginal density functions of V and W are both uniform: 1 fv(v) = fw(w) = 60 , 0 '.S V < 60, 0 '.S W < 60. i) Briefly explain ( one sentence only is required) why the joint density function of arrival times of Victor and Wilfred is 1 fv,w( v, w) = 3600' 0 '.S V < 60, 0 '.S W < 60. ii) Using fv,w(v, w), or otherwise, determine the probability that Victor arrives before Wilfred, i.e find P(V < W). iii) Let T = W - V be the time (in minutes) between Victor's arrival and Wilfred's arrival. A) Determine the cumulative distribution function Fy(t). B) Hence, or otherwise, determine the density function jy(t). Please see over ... Semester 1 2016 MATH2801 Page 4 3. [20 marks] Answer this question in a separate book a) Let X be a binomial variable, X rv Bin(n,p) for some 0 < p < 1, so P( X = x) = f x ( x) = (:) PX (1 - p r-x for x = 0, 1, 2, ... , n with JE(X) = np and Var(X) = np(l - p). i) Prove that X has moment generating function ii) Let Y = 2X. A) Determine my(u), the moment generating function for Y. B) Determine fy(y), the probability function for Y. C) Does Y have a binomial distribution for some suitable value of the parameters? Give a reason for your answer. b) A random sample of 100 postal employees found that the average time the employees had worked at the postal service was ?f = 7 years with a (sample) standard deviation of s = 2 years. Does this provide convincing evidence that the mean time of employment (in years) has changed from the value of 7.5 that was true 20 years ago? Carry out an appropriate hypothesis test to answer this question by an- swering the following: i) State the null and alternative hypotheses. ii) State any assumption(s) you are making in order to carry out this hypothesis test. State what you could do, if anything, to check the plausibility of your assumption( s). iii) State the formula for the test statistic and the null distribution. iv) Compute the observed value of the test statistic. v) Give an expression for the P-value, and then estimate it as best you can from the tables provided. vi) Write a relevant conclusion concerning the mean time of employment of postal employees. Please see over ... Semester 1 2016 MATH2801 Page 5 4. [20 marks] Answer this question in a separate book We wish to estimate p, the chance of obtaining a head when a particular coin is flipped. To do so, the coin is randomly flipped many times. For i = 1, ... , let Xi = 1 if a head is obtained on the ith flip and Xi = 0 otherwise. The Xi are i.i.d. Bernoulli(p) random variables for some 0 < p < l and the probability function can be given by P(Xi = x) = f(x) = {P 1-p or equivalently by for X = l for X = 0 for X = 0, 1. Suppose the coin is to be flipped n times, resulting in observations x1, x2, . .. , Xn, The sample proportion of heads can be computed by p = I::~o Xi = x. n a) Determine the method of moments estimator of p. Show that it is given by the sample proportion p. b) Determine the maximum likelihood estimator of p. Show that it is given by the sample proportion p. c) Determine the Fisher information for p. d) i) Write down the large sample approximation to the distribution of p. ii) Use the delta method to deduce the large sample approximation to the distribution of 1 fJ .. -p e) Assume now that the coin is flipped n = 100 times, and that a head is obtained 75 times in the 100 flips of the coin. Compute an (approximate) 95% confidence interval for p. Semester 1 2016 MATH2801 Page 6 Table of some common distributions X mass/ density domain mean variance mgf function mx(u) = E(euX) Bernoulli P(X = 1) = p {0,1} p pq q+peu P(X = 0) = q = 1 - p Binomial Bin(n,p) G)Pk(l - p)n-k k = 0,1, ... ,n np np(l - p) (1-p+peur Geometric p(l-p)k-1 k = l,2, ... l (1-p) p p ~ e-"-l+p Poisson e->->.k /kl k = 0,1, … ,\ ,\ exp{,\(eu -1)} Uniform (b – a)-1 xE(a,b) ½(a+b) f2(b-a) 2 ebu_eau u(b-a) Exponential 1 _],_ X xE(0,oo) f3 (32 1 73e f3 1-,Bu Normal N(µ,a-2) _l_ exp { – (x-µ)2 } v2-1ra2 ~ x E {-oo, oo} µ (T2 eµu+½CT2u2 Gamma (a,/3) e-xff3xa.-l xE(0,oo) Ol/3 a/32 ( 1-1/lu r r(a)/l” Beta /3( a, /3) r(a+/l) a-1c1- )/l-1 XE [0,1) “‘ a,B r(a)r(,B)x x a+,B (a+,B)2(a+/J+l) Some formulae Let X1, … , Xn be independent N(µ, cr2) variables. Then X and S2 are independent, and (n-l)S2 2 —- ,.___, Xn-1 · cr2 The Delta Method Let 01 , 02 , … be a sequence of estimators of 0 such that en -0 v cr/yn,-+ N(O, 1). Suppose the function g is differentiable at 0 and g’ ( 0) -/= 0. Then g(0n) – g(0) !!._,, N(O, 1). g'(0)cr I vn Semester 1 2016 MATH2801 Page 7 t distribution critical values Key: Table entry for p and C is the critical value t* with probability p lying to its right and probability C lying between -t* and t*. Upper tail probability p df .25 .20 .15 .10 .05 .025 .02 .01 .005 .0025 .001 1 1.000 1.376 1.963 3.078 6.314 12.71 15.89 31.82 63.66 127.3 318.3 2 0.816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9.925 14.09 22.33 3 0.765 0.978 1.250 1.638 2.353 3.182 3.482 4.541 5.841 7.453 10.21 4 0.741 0.941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 5.598 7.173 5 0.727 0.920 1.156 1.476 2.015 2.571 2.757 3.365 4.032 4.773 5.893 6 0.718 0.906 1.134 1.440 1.943 2.447 2.612 3.143 3.707 4.317 5.208 7 0.711 0.896 1.119 1.415 1.895 2.365 2.517 2.998 3.499 4.029 4.785 8 0.706 0.889 1.108 1.397 1.860 2.306 2.449 2.896 3.355 3.833 4.501 9 0.703 0.883 1.100 1.383 1.833 2.262 2.398 2.821 3.250 3.690 4.297 10 0.700 0.879 1.093 1.372 1.812 2.228 2.359 2.764 3.169 3.581 4.144 11 0.697 0.876 1.088 1.363 1.796 2.201 2.328 2.718 3.106 3.497 4.025 12 0.695 0.873 1.083 1.356 1.782 2.179 2.303 2.681 3.055 3.428 3.930 13 0.694 0.870 1.079 1.350 1.771 2.160 2.282 2.650 3.012 3.372 3.852 14 0.692 0.868 1.076 1.345 1.761 2.145 2.264 2.624 2.977 3.326 3.787 15 0.691 0.866 1.074 1.341 1.753 2.131 2.249 2.602 2.947 3.286 3.733 16 0.690 0.865 1.071 1.337 1.746 2.120 2.235 2.583 2.921 3.252 3.686 17 0.689 0.863 1.069 1.333 1.740 2.110 2.224 2.567 2.898 3.222 3.646 18 0.688 0.862 1.067 1.330 1.734 2.101 2.214 2.552 2.878 3.197 3.610 19 0.688 0.861 1.066 1.328 1.729 2.093 2.205 2.539 2.861 3.174 3.579 20 0.687 0.860 1.064 1.325 1.725 2.086 2.197 2.528 2.845 3.153 3.552 21 0.686 0.859 1.063 1.323 1.721 2.080 2.189 2.518 2.831 3.135 3.527 22 0.686 0.858 1.061 1.321 1.717 2.074 2.183 2.508 2.819 3.119 3.505 23 0.685 0.858 1.060 1.319 1.714 2.069 2.177 2.500 2.807 3.104 3.485 24 0.685 0.857 1.059 1.318 1.711 2.064 2.172 2.492 2.797 3.091 3.467 25 0.684 0.856 1.058 1.316 1.708 2.060 2.167 2.485 2.787 3.078 3.450 26 0.684 0.856 1.058 1.315 1.706 2.056 2.162 2.479 2.779 3.067 3.435 27 0.684 0.855 1.057 1.314 1.703 2.052 2.158 2.473 2.771 3.057 3.421 28 0.683 0.855 1.056 1.313 1.701 2.048 2.154 2.467 2.763 3.047 3.408 29 0.683 0.854 1.055 1.311 1.699 2.045 2.150 2.462 2.756 3.038 3.396 30 0.683 0.854 1.055 1.310 1.697 2.042 2.147 2.457 2.750 3.030 3.385 40 0.681 0.851 1.050 1.303 1.684 2.021 2.123 2.423 2.704 2.971 3.307 50 0.679 0.849 1.047 1.299 1.676 2.009 2.109 2.403 2.678 2.937 3.261 60 0.679 0.848 1.045 1.296 1.671 2.000 2.099 2.390 2.660 2.915 3.232 80 0.678 0.846 1.043 1.292 1.664 1.990 2.088 2.374 2.639 2.887 3.195 100 0.677 0.845 1.042 1.290 1.660 1.984 2.081 2.364 2.626 2.871 3.174 1000 0.675 0.842 1.037 1.282 1.646 1.962 2.056 2.330 2.581 2.813 3.098 .50 .60 .70 0.80 .90 .95 .96 .98 .99 .995 .998 Probability C .0005 636.6 31.60 12.92 8.610 6.869 5.959 5.408 5.041 4.781 4.587 4.437 4.318 4.221 4.140 4.073 4.015 3.965 3.922 3.883 3.850 3.819 3.792 3.768 3.745 3.725 3.707 3 .. 690 3.674 3.659 3.646 3.551 3.496 3.460 3.416 3.390 3.300 .999 Semester 1 2016 MATH2801 Page 8 Standard normal probabilities Key: Table entry for z is the area under the standard normal curve to the left of z . z . 00 .01 .02 .03 .04 .05 .06 .07 .08 .09 -2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 -2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019 -2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026 -2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036 -2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048 -2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064 -2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084 -2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110 -2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143 -2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183 -1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233 -1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294 -1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 -1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 -1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559 -1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681 -1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 -1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985 -1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170 -1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379 -0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611 -0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867 -0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148 -0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 -0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776 -0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121 -0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483 -0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859 -0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247 -0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641 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 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 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 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 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 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9641 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 0.9817 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 0.9911 0.9913 0.9916 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 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 0.9981 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986

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