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IIT JAM 2012 : Mathematical Statistics

35 pages, 44 questions, 17 questions with responses, 17 total responses,

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A 2012 MS 2012 MS Test Paper Code: MS Time: 3 Hours Maximum Marks: 300 INSTRUCTIONS 1. This question-cum-answer booklet has 32 pages and has 25 questions. Please ensure that the copy of the question-cum-answer booklet you have received contains all the questions. 2. Write your Registration Number, Name and the name of the Test Centre in the appropriate space provided on the right side. 3. Write the answers to the objective questions against each Question No. in the Answer Table for Objective Questions, provided on Page No. 7. Do not write anything else on this page. 4. Each objective question has 4 choices for its answer: (A), (B), (C) and (D). Only ONE of them is the correct answer. There will be negative marking for wrong answers to objective questions. The following marking scheme for objective questions shall be used: (a) For each correct answer, you will be awarded 6 (Six) marks. (b) For each wrong answer, you will be awarded - 2 (Negative two) marks. (c) Multiple answers to a question will be treated as a wrong answer. (d) For each un-attempted question, you will be awarded 0 (Zero) mark. (e) Negative marks for objective part will be carried over to total marks. 5. Answer the subjective question only in the space provided after each question. 6. Do not write more than one answer for the same question. In case you attempt a subjective question more than once, please cancel the answer(s) you consider wrong. Otherwise, the answer appearing last only will be evaluated. 7. All answers must be written in blue/black/blueblack ink only. Sketch pen, pencil or ink of any other colour should not be used. 8. All rough work should be done in the space provided and scored out finally. 9. No supplementary sheets will be provided to the candidates. 10. Clip board, log tables, slide rule, calculator, cellular phone and electronic gadgets in any form are NOT allowed. 11. The question-cum-answer booklet must be returned in its entirety to the Invigilator before leaving the examination hall. Do not remove any page from this booklet. 12. Refer to special instructions/useful data on the reverse. MS- i / 32 READ INSTRUCTIONS ON THE LEFT SIDE OF THIS PAGE CAREFULLY REGISTRATION NUMBER Name: Test Centre: Do not write your Registration Number or Name anywhere else in this question-cum-answer booklet. I have read all the instructions and shall abide by them. ... Signature of the Candidate I have verified the information filled by the Candidate above. ... Signature of the Invigilator Special Instructions/ Useful Data 1. : Set of all real numbers. 2. i.i.d. : independent and identically distributed. 3. N ( , 2 ) : Normal distribution with mean and variance 2 0 . 4. For a fixed 0 , X ~ Exp( ) means that the probability density function of random variable X is e x if x 0, f ( x; ) otherwise. 0 5. U (a, b) : Uniform distribution on (a, b) , a b . 6. B (n, p ) : Binomial distribution with parameters n {1, 2,...} and p (0,1) . 7. E ( X ) : Expectation of X. 8. Fm ,n : F distribution with m and n degrees of freedom. 1 n xi is the sample mean based on ( x1 ,..., xn ) . n i 1 10. = P [type I error] and = P [type II error] 11. H 0 : Null Hypothesis, H1 : Alternative Hypothesis 9. x Useful data z 2 x 1 ( z) e 2 dx , where ( z ) is cumulative distribution 2 function of N (0, 1) . (1.28) 0.900 , (1.65) 0.950 , (1.96) 0.975 , (2.33) 0.990 , (2.58) 0.995 . MS- ii / 32 Q.1 IMPORTANT NOTE FOR CANDIDATES Questions 1-15 (objective questions) carry six marks each and questions 16-25 (subjective questions) carry twenty one marks each. Write the answers to the objective questions in the Answer Table for Objective Questions provided on page 7 only. 2 1 0 An eigenvector of the matrix M 0 2 1 is 0 0 2 1 (A) 0 0 Q.2 0 (B) 1 0 (B) 4 1 (A) I y xy dy dx. The change of order of integration in the integral gives I as 2 xy dx dy 0 1 (C) I 0 1 y 0 0 2 y 1 0 xy dx dy 1 0 xy dx dy xy dx dy. 2 y 0 2 y 0 xy dx dy. 2 0 1 (D) I 0 2 y 0 2 y 1 0 (B) I (D) 12 x2 1 Let I (C) 8 2 x 0 Q.4 2 (D) 2 2 The volume of the solid of revolution generated by revolving the area bounded by the curve y x and the straight lines x 4 and y 0 about the x axis, is (A) 2 Q.3 0 (C) 0 1 xy dx dy. 2 1 xy dx dy y 0 xy dx dy. 1 2 k Let L lim n f f f k f (0) , where k is a positive integer. If n n n n f ( x) sin x, then L is equal to (A) (k 1)(k 2) 6 (B) (k 1)(k 2) 2 MS- 1 / 32 (C) k (k 1) 2 (D) k (k 1) Q.5 1 x 2 y 2 sin 2 2 Let f ( x, y ) x y 0 Then at the point (0, 0), if ( x, y ) (0, 0) if ( x, y ) (0, 0). f f exist. and x y f f (B) f is continuous and do not exist. and x y f f (C) f is not continuous and exist. and x y f f (D) f is not continuous and do not exist. and x y (A) f is continuous and Q.6 Let an be a real sequence converging to a, where a 0. Then (B) a an 1 (A) n n converges but an diverges but (D) Both a 1 a 1 Q.7 and an converge. an diverge. n 1 n 1 (B) 0.973 (C) 0.027 (D) 0.27 A person makes repeated attempts to destroy a target. Attempts are made independent of each other. The probability of destroying the target in any attempt is 0.8. Given that he fails to destroy the target in the first five attempts, the probability that the target is destroyed in the 8th attempt is (A) 0.128 Q.9 n and converges. 1 A four digit number is chosen at random. The probability that there are exactly two zeros in that number is (A) 0.73 Q.8 n an n 1 (C) Both diverges. 1 (B) 0.032 (C) 0.160 (D) 0.064 Let the random variable X B(5, p) such that P ( X 2 ) 2 P( X 3 ) . Then the variance of X is (A) 10 3 (B) 10 9 (C) MS- 2 / 32 5 3 (D) 5 9 Q.10 8 Let X 1 , , X 8 be i.i.d. N (0, 2 ) random variables. Further, let U X 1 X 2 and V X i . i 1 The correlation coefficient between U and V is (A) Q.11 1 8 (C) 3 4 (D) 1 2 Let X ~ F8,15 and Y ~ F15,8 . If P ( X 4 ) 0.01 and P(Y k ) 0.01 , then the value of k is (A) 0.025 Q.12 1 4 (B) (B) 0.25 (C) 2 (D) 4 n Let X 1 , , X n be i.i.d. Exp(1) random variables and S n X i . Using the central limit i 1 theorem, the value of lim P( S n n) is n (A) 0 Q.13 (B) 1 3 (C) 1 2 (D) 1 Let the random variable X U (5,5 ) . Based on a random sample of size 1, say X 1 , the unbiased estimator of 2 is (A) 3( X 1 5) 2 Q.14 X 12 5 (B) 12 (C) 3( X 1 5) 2 X 12 5 (D) 12 Let X 1 , , X n be a random sample of size n from N ( ,16) population. If a 95% confidence interval for is , then the value of n is X 0.98, X 0.98 (A) 4 Q.15 (B) 16 (C) 32 (D) 64 A coin is tossed 4 times and p is the probability of getting head in a single trial. Let S be the number of head(s) obtained. It is decided to test 1 1 H 0 : p against H1 : p , using the decision rule : Reject H 0 if S is 0 or 4. The 2 2 3 probabilities of Type I error ( ), and Type II error ( ) when p = , are 4 1 87 (A) , 4 128 1 87 (B) , 8 128 1 41 (C) , 8 256 1 41 (D) , 4 256 MS- 3 / 32 Space for rough work MS- 4 / 32 Space for rough work MS- 5 / 32 Space for rough work MS- 6 / 32 Answer Table for Objective Questions Write the Code of your chosen answer only in the Answer column against each Question Number. Do not write anything else on this page. Question Answer Number Do not write in this column 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 FOR EVALUATION ONLY Number of Correct Answers Marks (+) Number of Incorrect Answers Marks ( ) Total Marks in Questions 1-15 MS- 7 / 32 ( ) Q.16 (a) Find the value(s) of for which the following system of linear equations 1 1 x 1 1 1 y 1 1 1 z 1 (i) has a unique solution, (ii) has infinitely many solutions, (iii) has no solution. (b) Let a1 2, b1 1 and for n 1, an 1 (9) an bn 2anbn , bn 1 . Show that 2 an bn (i) bn an for all n, (ii) bn 1 bn for all n, (iii) the sequences an and bn converge to the same limit MS- 8 / 32 2. (12) MS- 9 / 32 Q.17 (a) Solve: x 2 y 3 xy dy dx . (9) (b) Find the general solution of the differential equation d . D 2 4 D 4 y x sin 2 x, where D dx MS-10 / 32 (12) MS-11 / 32 Q.18 (a) (b) Find all the critical points of the function f ( x, y ) x 3 y 3 3 x y and examine those points for local maxima and local minima. (9) If f is a continuous real-valued function on [0,1] , show that there exists a 1 point c (0,1) such that 1 0 c x f ( x) dx f ( x) dx . MS-12 / 32 (12) MS-13 / 32 Q.19 (a) z 4 x 2 z y 4 z x2 Evaluate the triple integral: z 0 (b) x 0 dy dx dz . y 0 1 2 0 5 1 2 1 Let M 0 2 1 . If M 4 I k M 4 M , where I is the identity matrix 1 0 1 of order 3, find the value of k. Hence or otherwise, solve the system of x 1 equations: M y 0 . z 0 MS-14 / 32 (9) (12) MS-15 / 32 Q.20 (a) Let N be a random variable representing the number of fair dice thrown with 1 probability mass function P ( N i ) i , i 1, 2, . Let S be the sum of the 2 numbers appearing on the faces of the dice. Given that S = 3, what is the probability that 2 dice were thrown ? (b) Let X N (0,1) and Y X X . Find E (Y 3 ) . MS-16 / 32 (9) (12) MS-17 / 32 2 Q.21 Let Y N ( y , y ) and Y ln X . (a) Find the probability density function of the random variable X and the median of X. (b) Find the maximum likelihood estimator of the median of the random variable X based on a random sample of size n. MS-18 / 32 (9) (12) MS-19 / 32 Q.22 (a) A random variable X has probability density function f ( x) x e If E ( X ) 2 2 x2 , x 0, 0, 0 . , determine and . (b) Let X and Y be two random variables with joint probability density function e y if 0 x y , f ( x, y ) otherwise. 0 (i) Find the marginal density functions of X and Y . (ii) Examine whether X and Y are independent. (iii) Find Cov( X , Y ) . MS-20 / 32 (9) (12) MS-21 / 32 Q.23 (a) 1 Let X 1 , , X n be a random sample from Exp population. Obtain the Cramer Rao lower bound for the variance of an unbiased estimator of 2 . (9) (b) Let X 1 , , X n (n > 4) be a random sample from a population with mean and variance 2 . Consider the following estimators of 1 n 1 3 1 ( X 2 X n 1 ) X n . U Xi , V X1 8 4(n 2) 8 n i 1 (i) Examine whether the estimators U and V are unbiased. (ii) Examine whether the estimators U and V are consistent. (iii) Which of these two estimators is more efficient? Justify your answer. MS-22 / 32 (12) MS-23 / 32 Q.24 Let X 1 , , X n be a random sample from a Bernoulli population with parameter p. (a) (i) Find a sufficient statistic for p. p(1 p) given by (ii) Consider an estimator U ( X 1 , X 2 ) of n 1 if X 1 X 2 1, U ( X 1 , X 2 ) 2n otherwise. 0 Examine whether U ( X 1 , X 2 ) is an unbiased estimator. (9) (b) Using the results obtained in (a) above and Rao Blackwell theorem, find the p(1 p) uniformly minimum variance unbiased estimator (UMVUE) of . n MS-24 / 32 (12) MS-25 / 32 Q.25 (a) Let X 1 , , X n be a random sample from the population having probability density function 2 x x2 2 if x 0 f ( x, ) 2 e 0 otherwise. Obtain the most powerful test for testing H 0 : 0 against H1 : 1 ( 1 0 ). (b) Let X 1 , , X n be a random sample of size n from N ( ,1) population. To test H 0 : 5 against H1 : 4 , the decision rule is : Reject H 0 if x c . If 0.05 and 0.10 , determine n (rounded off to an integer) and hence c. MS-26 / 32 (9) (12) MS-27 / 32 Space for rough work MS-28 / 32 Space for rough work MS-29 / 32 Space for rough work MS-30 / 32 Space for rough work MS-31 / 32 Space for rough work MS-32 / 32 2012 MS Objective Part ( Question Number 1 15) Total Marks Signature Subjective Part Question Number 16 Question Number 21 Marks 17 22 18 23 19 24 20 25 Total Marks in Subjective Part Total (Objective Part) : Total (Subjective Part) : Grand Total : Total Marks (in words) : Signature of Examiner(s) : Signature of Head Examiner(s) : Signature of Scrutinizer : Signature of Chief Scrutinizer : Signature of Coordinating Head Examiner : MS-iii / 32 Marks

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