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IIT JAM 2011 : Mathematical Statistics

35 pages, 33 questions, 15 questions with responses, 17 total responses,

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A 2011 MS 2011 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) mark. (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. : Set of all rational numbers. 3. xT : Transpose of a column vector x. 4. i.i.d.: independent and identically distributed. 5. N ( , 2 ) : Normal distribution with mean and variance 2 > 0. 6. For a fixed > 0, X is Exp ( ) random variable means that the probability density function of X is e x , if x > 0, f (x | ) = otherwise. 0, 7. U ( a, b ) : Continuous Uniform distribution on ( a, b ) , < a < b < . 8. B ( n, p ) : Binomial distribution with parameters n {1, 2, } and p ( 0,1) . 9. E ( X ) : Expectation of a random variable X . 10. Based on the observations ( x1 , , xn ) , x= 1 n n x i =1 i is the sample mean 1 n 2 ( xi x ) is the sample variance. n 1 i = 1 11. tn : Central Student s t random variable with n degrees of freedom. 2 = and sx 12. P ( t5 2.57 ) = 0.975, P ( t5 2.01) = 0.95, P ( t4 2.78 ) = 0.975, P ( t4 2.13) = 0.95. 2 13. n : Central Chi-square random variable with n degrees of freedom. 2 2 2 14. P ( 20 > 10.85 ) = 0.95, P ( 10 > 3.94 ) = 0.95, P ( 20 > 21.7 ) = 0.36, 2 P ( 10 > 7.88 ) = 0.69. 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. Let the function f : [ 0, ) (B) 4 e 2 (A) e 1 Q.2 (C) 9 e 3 (D) 16 e 4 1 8 An eigen-vector of the matrix is 0 1 (A) (1, 2 ) (B) ( 5, 0 ) T Q.3 be given by f ( x ) = x 2 e x . Then the maximum value of f is Define f , g : T (C) ( 0, 2 ) (D) (1,1) T T by if x , x, f ( x) = sin ( x ) , if x x sin ( x ) sin (1 x ) , if x 0, and g ( x ) = if x = 0. 0, At x = 0, (A) both f and g are differentiable. (B) f is differentiable but g is NOT differentiable. (C) g is differentiable but f is NOT differentiable. (D) neither f nor g is differentiable. Q.4 Consider the series S1 and S 2 given by: S1 : n =1 n2 + n + 1 n ( n + 1) and Then (A) both S1 and S 2 converge. (B) S1 converges and S 2 diverges. (C) S 2 converges and S1 diverges. (D) both S1 and S 2 diverge. Q.5 The equation x13 e x + x sin ( x ) = 0 has (A) no real root. (B) more than two real roots. (C) exactly two real roots. (D) exactly one real root. MS- 1 / 32 S2 : n =1 n2 + 1 . n 2 ( n + 1) Q.6 Let D be the triangle bounded by the y axis, the line 2 y = and the line y = x. Then the cos ( y ) value of the integral dx dy is y D 1 2 3 (C) 2 (A) Q.7 (B) 1 (D) 2 Let X be a random sample of size one from U ( , + 1) distribution, . For testing H 0 : = 1 against H1 : = 2 , the critical region { x : x > 1} has (A) power = 1 and size = 1. (B) power = 0 and size = 1. (C) power = 1 2 and size = 1. (D) power = 1 and size = 0. Q.8 Let X 1 ,..., X n be i.i.d. B (1, ) random variables, 0 < < 1. Then, as an estimator of , n T ( X 1 , , X n ) = X i =1 i + n 2 n+ n is (A) both consistent and unbiased. (B) consistent but NOT unbiased. (C) unbiased but NOT consistent. (D) neither unbiased nor consistent. Q.9 Let X 1 , X 2 , X 3 be i.i.d. N ( 0, 2 ) random variables, > 0. Then the value of k for which 3 the estimator k X i is an unbiased estimator of is i =1 (A) 1 3 (B) 2 9 (C) MS- 2 / 32 18 (D) 2 3 Q.10 Let the random variables X and Y have the joint probability mass function y x y 2x 2 x 3 1 P ( X = x, Y = y ) = e y = 0, 1, , x; x = 0, 1, 2, . ; x! y 4 4 Then E (Y ) = 1 2 3 (C) 2 (B) 1 (A) Q.11 (D) 2 Let X 1 and X 2 be i.i.d. Poisson random variables with mean 1. Then P ( max ( X 1 , X 2 ) > 1) = (A) 1 e 2 Q.12 (B) 1 2 e 2 (C) 1 3e 2 A fair die is rolled 3 times. The conditional probability of 6 appearing exactly once, given that it appeared at least once, equals 2 2 1 5 3 6 6 (A) 3 5 1 6 2 1 5 3 6 6 (C) 3 5 1 6 Q.13 1 5 6 6 (B) 3 5 1 6 2 1 5 6 6 (D) 3 5 1 6 2 Let X ~ B ( 2, 1 2 ) . Then E 1+ X (A) Q.14 (D) 1 4 e 2 7 6 = (B) 1 (C) 6 7 (D) 2 3 The moment generating function of an integer valued random variable X is given by 1 M X ( t ) = ( 2 + e t + 4 e 2 t + 3 e 3 t ) e t . 10 Then P ( 2 X + 5 < 7 ) = (A) 3 10 (B) 7 10 (C) 1 MS- 3 / 32 (D) 4 10 Q.15 Let X 1 and X 2 be i.i.d. Exp ( 3) random variables. Then P ( X 1 + X 2 > 1) = (A) 2 e 3 (B) 3 e 3 (C) 4 e 3 Space for rough work MS- 4 / 32 (D) 5e 3 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) Student population of a university has 30% Asian, 40% American, 20% European and 10% African students. It is known that 40% of all Asian students, 50% of all American students, 60% of all European students and 20% of all African students are girls. Find the probability that a girl chosen at random from the university is an (12) Asian. (b) Let A1 , A2 and A3 be pairwise independent events with P ( Ai ) = 1 , i = 1, 2, 3. 2 Suppose that A3 and A1 A2 are independent. Find the value of P ( A1 A2 A3 ) . MS- 8 / 32 (9) MS- 9 / 32 Q.17 Let X be a random variable with cumulative distribution function 0, x2 + 1 , 4 1 F ( x) = x + , 8 x +1 2 , 1, if x < 0, 1 if 0 x < , 2 1 3 if x < , 2 4 3 if x < 1, 4 if x 1. 1 3 1 Find the values of P 0 X < , P X and P X = . 4 4 2 MS-10 / 32 (21) MS-11 / 32 Q.18 (a) Let X be a continuous random variable with probability density function 1 x 1 e ; 2 f ( x) = < x < . Find the value of P (1 < X < 2 ) . (b) Let X and Y be i.i.d. U (0,1) random variables. Find the value of 1 P X 2 + Y 2 1 . 4 MS-12 / 32 (12) (9) MS-13 / 32 Q.19 (a) Let the random variables X 1 and X 2 have joint probability density function x1 e x 1 x 2 , if 1 < x1 < 3, x2 > 0, f ( x1 , x2 ) = 2 0, otherwise. Find the covariance between X 1 and X 2 . (12) (b) Let X 1 , , X 100 be i.i.d. U ( 0.5, 0.5) random variables and let T = X 1 + Using Chebyshev s inequality show that P (T 2 25 ) MS-14 / 32 1 . 3 + X 100 . (9) MS-15 / 32 Q.20 (a) Let X 1 , , X n be a random sample from a population having a probability density function 4 4 3 x x e , if x > 0, f ( x | ) = 0, otherwise, where > 0. Find the uniformly minimum variance unbiased estimator of . (9) 2 (b) Let x = 9 and sx = 6 be the sample mean and sample variance, respectively, based 2 = 4 be the on a random sample of size 3 from N ( 1 , 2 ) . Also let y = 7 and s y sample mean and sample variance, respectively, based on a random sample of size 3 from N ( 2 , 2 2 ) , where 1 , 2 and 2 > 0 are unknown. Find a 95% confidence interval for 1 2 . (12) MS-16 / 32 MS-17 / 32 Q.21 Let X 1 , , X 10 be a random sample of size 10 from a population having a probability density function , if x > 1, f ( x | ) = x +1 otherwise, 0, where > 0. For testing H 0 : = 2 against H1 : = 4 at the level of significance = 0.05 , find the most powerful test. Also find the power of this test. (21) MS-18 / 32 MS-19 / 32 Q.22 Let f : [ 1,1] be a continuous function. (a) Show that 1 1 f f is a convergent series. n +1 n = 1 n (9) (b) Further, if f is differentiable on ( 0,1) and f ( x ) < 1 for all x ( 0,1) , then show that 1 1 f f n n +1 n =1 is a convergent series. (12) MS-20 / 32 MS-21 / 32 Q.23 (a) Let f : [ 0,1] 1 be a continuous function such that there exists a point c ( 0, 1) such that f ( c ) = 3 c 2 . f ( t ) dt = 1. Then show that 0 (12) (b) Find the general solution of d2y dy 3 + 2 y = e x, 2 dx dx where it is given that y = x e x is a particular solution. MS-22 / 32 (9) MS-23 / 32 Q.24 (a) Let f : be defined by f ( x, y ) = x 2 + x y + y 2 x 100. Find the points of local maximum and local minimum, if any, of f . (b) Find lim n 43 n sin ( n ) . 3 4n (12) (9) MS-24 / 32 MS-25 / 32 Q.25 (a) Consider the following matrix 1 1 66 6 11 1 4 2 A= . 66 6 11 1 7 66 6 11 Find and so that A becomes an orthogonal matrix. Using these values of and , solve the system of equations x 1 A y = 0 . z 1 1 0 (b) Let A = 1 1 5 7 9 1 3 5 . Find so that the rank of A is two. 6 10 14 4 4 MS-26 / 32 (12) (9) 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 2011 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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