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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1302 PROBABILITY AND STATISTICS II June 1, 2007 Time: 2:30 p.m. - 4:30 p.m. Candidates taking examinations 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 purposes of calculation. It 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 ALL FOUR questions. Marks are shown in square brackets. 1. John and Mary play a game as follows. John selects a positive integer 0 E {1, 2, . . at random and writes it down on a piece of paper. Mary has two chances to ask John what the number 0 is. Each time John tosses a biased coin secretly, and reports to Mary the number 6 — 1 if a head comes up and the number 0 + 1 if a tail comes up. Mary has to guess the true value of 0 based on the two numbers reported by John after two independent tosses of the coin. It is known to Mary that the coin has a probability 1/3 of showing a head. Let X and Y be the two numbers which John reported. (a) What is the expected value of X? [3 marks] X + Y 1 (b) Define T — 2 — (i) Show that T is an unbiased estimator of 9. [4 marks] 4 (ii) Show that T has variance [4 marks] (c) Define X + Y if X ¢ Y, S = 1 X—g ifX=Y. S&AS: STAT1302 Probability and Statistics II 2 (i) Show that 1P(S=6)=§, 1P (see—g) 2%, r (5:0+§) :3. [6marks] (ii) Calculate the bias of S as an estimator of 6. Is S unbiased? [4 marks] (iii) Calculate the mean squared error of S as an estimator of 6. [4 marks] (d) Which of the estimators S and T should Mary use if she wants to minimize the mean squared error of her estimation? [3 marks] 2. The numbers of male and female customers who visited a company on n suc— cessive days were recorded as (X1,Y1),(X2,Y2),. . . , (XmYn), such that there were X1- male and Y,- female visitors on the 2th day. Assume that for each i, Xi and Y; follow Poisson distributions with means A and ,8)‘ reSpectively, for A, 5 > 0, and that the 271 observations are independent. It is known that the Poisson distribution with mean 6 > 0 has the mass function 90 6"- 17 f(ac[6)= x! , x=0,1,2,.... (a) Write down the likelihood function of (A, 5) based on the observations (X1, Y1), . . . , (Xn, Y”). [4 marks] (b) If it is known that ,8 = 1, show that the statistic T = 2m + K) i=1 is sufficient for A. [4 marks] (c) If both A and 5 are unknown, show that the statistic is sufficient for (A, fl). [4 marks] (d) Suppose that fl = A but the value of /\ is unknown. (i) Show that the statistic T : im + 2Yi) i=1 is sufficient for A. [4 marks] (ii) Show that the Fisher information based on the 271 observations is i()\)=n<4+§). [6 marks] S&AS: STAT1302 Probability and Statistics II 3 (iii) Show that the maximum likelihood estimator of A is i T 3\= — 16+2n 1 71. Z, Where T = + 236). [4 marks] (iv) If n is large, the maximum likelihood estimator 5x has an approx— imately normal distribution. State the mean and variance of this normal distribution. [4 marks] (v) It is observed that the total numbers of male and female visitors on n = 20 successive days are ‘20 20 in=40 and Zia-=10 i=1 i=1 respectively. Calculate the maximum likelihood estimate of A and its standard error. [4 marks] 3. Let X and Y be independent random variables having the density functions 9(1+ :c)—1_9, :1: > 0, 0, m S 0; 06—9”, y > O, 0, y S 0, f(93|9) ={ and 9(yl9) = { respectively, for a common unknown parameter 0 > 0. It is known that T = (1 + X )‘he‘y has the cumulative distribution function 0 iftgo, 1m" 3 t | 6) = t9(1—01nt) ifte (0,1), 1 iftZl. Based on one observation of (X, Y), we wish to test H026§c against H1:6>C, for some specified constant c > 0. (a) Write down the likelihood function of 0 based on (X , Y). [3 marks] (b) Show that the given model has monotone likelihood ratio in the statistic T = (1 + X)"le_y. [5 marks] (0) Describe the critical region of size 0.05 for the uniformly most powerful test of H0 against H1. You are not required to derive the critical value explicitly. [5 marks] (d) It is observed that X =0.25 and Y: ln . S&AS: STAT1302 Probability and Statistics II 4 (i) Calculate the value of T. [2 marks] (ii) Show that the p-value for testing Hozagl against H1:0>1 is approximately 0.04348. Would you reject H0 at the 5% significance level? [5 marks] (iii) Appendix I shows a plot of the function t(c) against c, where t(c) satisfies t(c)c(1 — clnt(c)) = 0.95. For what values of 0 would you reject H0 : 6 _<_ c in favour of H1 : 0 > c at the 5% significance level? Hence deduce a 95% lower confidence bound for 0. [6 marks] 4. A random sample {X 1, . . . , X 3} is drawn from the normal distribution N (a, 0?). Define 8 8 X = ZXi/B and S = Z(X,- — )2)? i=1 i=1 It is known that 8/02 has a chi-squared distribution. The statistic S is calculated to be 2.5 based on a realisation of the above normal sample. (a) Find the value of c which satisfies lP’(S/02 2 c) = 0.95. [3 marks] , 5% lower and upper quantiles of X2 1 2 3 4 5 6 7 8 0.0039 0.1026 0.3518 0.7107 1.1455 1.6354 2.1674 2.7326 3.8415 5.9915 7.8147 9.4877 11.0705 12.5916 14.0671 15.5073 (b) Calculate a 95% upper confidence bound for 02. [3 marks] (c) Calculate a 95% lower confidence bound for 1/0. [3 marks] (d) Would you reject H0 : a = 0.2 in favour of H1 : a < 0.2 at the 5% significance level? [3 marks] t(c) S&AS: STAT1302 Probability and Statistics II 5 APPENDIX I Plot of t(c) against 0 0.96 0.90 0.84 0.78 0.72 0.66 0.60 0.54 0.48 0.42 0.36 0.30 0.24 0.18 0.12 0.06 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.5 3.8 ...
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This note was uploaded on 01/27/2011 for the course STAT 1302 taught by Professor Smslee during the Spring '10 term at HKU.

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dabbb - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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