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IIT JAM 2006 : Mathematical Statistics

9 pages, 39 questions, 6 questions with responses, 6 total responses,

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JAM 2006 MATHEMATICAL STATISTICS TEST PAPER 1 Special Instructions / Useful Data 1. For an event A, P ( A ) denotes the probability of the event A. 2. The complement of an event is denoted by putting a superscript c on the event, e.g. Ac denotes the complement of the event A. 3. For a random variable X , E ( X ) denotes the expectation of X and V ( X ) denotes its variance. ( ) 4. N , 2 denotes a normal distribution with mean and variance 2 . 5. Standard normal random variable is a random variable having a normal distribution with mean 0 and variance 1. 6. P ( Z > 1.96 ) = 0.025, P ( Z > 1.65 ) = 0.050, P ( Z > 0.675 ) = 0.250 and P ( Z > 2.33) = 0.010 , where Z is a standard normal random variable. 2 2 7. P 2 9.21 = 0.01, P 2 0.02 = 0.99, ( P( 2 4 ) ( ) 0.71) = 0.95 , P ( 11.07 ) = 0.05 2 5 ( ) P 32 11.34 = 0.01, ( ) ( ( ) 2 P 4 9.49 = 0.05, ) 2 2 and P 52 1.15 = 0.95, where P n c = , where n has a Chi-square distribution with n degrees of freedom. 8. n ! denotes the factorial of n. 9. The determinant of a square matrix A is denoted by | A | . 10. R: The set of all real numbers. 11. R : n-dimensional Euclidean space. 12. y and y denote the first and second derivatives respectively of the function y ( x) with respect to x. NOTE: This Question-cum-Answer book contains THREE sections, the Compulsory Section A, and the Optional Sections B and C. Attempt ALL questions in the compulsory section A. It has 15 objective type questions of six marks each and also nine subjective type questions of fifteen marks each. Optional Sections B and C have five subjective type questions of fifteen marks each. Candidates seeking admission to either of the two programmes, M.Sc. in Applied Statistics & Informatics at IIT Bombay and M.Sc. in Statistics & Informatics at IIT Kharagpur, are required to attempt ONLY Section B (Mathematics) from the Optional Sections. Candidates seeking admission to the programme, M.Sc. in Statistics at IIT Kanpur, are required to attempt ONLY Section C (Statistics) from the Optional Sections. You must therefore attempt either Optional Section B or Optional Section C depending upon the programme(s) you are seeking admission to, and accordingly tick one of the boxes given below. Optional Section Attempted B C The negative marks for the Objective type questions will be carried over to the total marks. Write the answers to the objective questions in the Answer Table for Objective Questions provided on page MS 11/63 only. 1 Compulsory Section A 1. If an > 0 for n 1 and lim n ( an ) 1n = L < 1, then which of the following series is not convergent? (A) n =1 (B) a n =1 (C) 2 n an 1 an n =1 (D) an an +1 n =1 2. Let E and F be two mutually disjoint events. Further, let E and F be independent of G. p = P ( E ) + P ( F ) and q = P (G ) , then P ( E F G ) is (A) 1 pq (B) q + p 2 (C) p + q 2 (D) p + q pq If 3. Let X be a continuous random variable with the probability density function symmetric about 0. If V ( X ) < , then which of the following statements is true? (A) E (| X |) = E ( X ) (B) V (| X |) = V ( X ) (C) V (| X |) < V ( X ) (D) V (| X |) > V ( X ) 4. Let f ( x) = x | x | + | x 1|, < x < . Which of the following statements is true? (A) f is not differentiable at x = 0 and x = 1. (B) f is differentiable at x = 0 but not differentiable at x = 1. (C) f is not differentiable at x = 0 but differentiable at x = 1. (D) f is differentiable at x = 0 and x = 1. 5. Let A x = b be a non-homogeneous system of linear equations. The augmented matrix [ A : b ] is given by %% % 1 1 2 1 1 1 2 3 1 0 . 0 3 1 0 1 2 Which of the following statements is true? (A) Rank of A is 3. (B) The system has no solution. (C) The system has unique solution. (D) The system has infinite number of solutions. 6. An archer makes 10 independent attempts at a target and his probability of hitting the target at each attempt 5 is . Then the conditional probability that his last two attempts are successful given that he has a total of 7 6 successful attempts is 1 (A) 5 5 7 (B) 15 25 (C) 36 7 8! 5 1 (D) 3! 5! 6 6 3 7. Let f ( x) = ( x 1)( x 2 ) ( x 3) ( x 4 ) ( x 5 ) , < x < . d f ( x) = 0 is exactly dx (D) 5 The number of distinct real roots of the equation (A) 2 (B) 3 (C) 4 8. Let f ( x) = k | x| (1 + | x |) 4 , < x < . Then the value of k for which f ( x) is a probability density function is 1 (A) 6 1 (B) 2 (C) 3 (D) 6 9. If M X ( t ) = e 3t +8t 2 is the moment generating function of a random variable X , then P ( 4.84 < X 9.60 ) is (A) equal to 0.700 (B) equal to 0.925 (C) equal to 0.975 (D) greater than 0.999 3 10. Let X be a binomial random variable with parameters n and p, where n is a positive integer and ( ) 0 p 1. If = P | X np | n , then which of the following statements holds true for all n and p? 1 4 1 1 < (B) 4 2 1 3 < < (C) 2 4 (A) 0 (D) 3 1 4 11. Let X 1 , X 2 ,..., X n be a random sample from a Bernoulli distribution with parameter p; 0 p 1. The n n + 2 X i bias of the estimator 1 n +1 1 (B) n+ n 1 (C) n +1 1 (D) n +1 (A) ( i =1 2 n+ n ) for estimating p is equal to 1 p 2 1 p 2 p 1 2 + p n 1 p 2 12. Let the joint probability density function of X and Y be e x , if 0 y x < , f ( x, y ) = otherwise. 0, Then E ( X ) is (A) (B) (C) (D) 0.5 1 2 6 4 13. Let f : be defined as tan t , t 0, f (t ) = t 1, t = 0. 1 Then the value of lim 2 x 0 x x3 f ( t ) dt x2 (A) is equal to 1 (B) is equal to 0 (C) is equal to 1 (D) does not exist 14. Let X and Y have the joint probability mass function; x 2 y +1 1 P ( X = x, Y = y ) = y + 2 , x, y = 0,1, 2,... . 2 ( y + 1) 2 y + 2 Then the marginal distribution of Y is 1 (A) Poisson with parameter = 4 1 (B) Poisson with parameter = 2 1 (C) Geometric with parameter p = 4 1 (D) Geometric with parameter p = 2 15. Let X 1 , X 2 and X 3 be a random sample from a N ( 3, 12 ) distribution. If X = S2 = ( 2 13 ( Xi X ) 2 i =1 13 X i and 3 i =1 denote the sample mean and the sample variance respectively, then ) P 1.65 < X 4.35, 0.12 < S 2 55.26 is (A) (B) (C) (D) 0.49 0.50 0.98 none of the above 5 16. (a) Let X 1 , X 2 , ... , X n be a random sample from an exponential distribution with the probability density function; e x , if x > 0, f (x; ) = otherwise, 0, where > 0. Obtain the maximum likelihood estimator of P ( X > 10 ) . 9 Marks (b) Let X 1 , X 2 , ... , X n be a random sample from a discrete distribution with the probability mass function given by 1 1 P ( X = 0) = ; P ( X = 1) = ; P ( X = 2 ) = , 0 1. 2 2 2 6 Marks Find the method of moments estimator for . 17. (a) Let A be a non-singular matrix of order n (n > 1), with | A | = k . If adj ( A) denotes the adjoint of the matrix A , find the value of | adj ( A) | . 6 Marks (b) Determine the values of a, b and c so that (1, 0, 1) and ( 0, 1, 1) are eigenvectors of the matrix, 2 1 1 a 3 2 . 3 b c 9 Marks 18. (a) Using Lagrange s mean value theorem, prove that b a b a , < tan 1 b tan 1 a < 2 1+ b 1 + a2 where 0 < tan 1 a < tan 1 b < . 6 Marks 2 (b) Find the area of the region in the first quadrant that is bounded by y = x , y = x 2 and the x axis . 9 Marks 19. Let X and Y have the joint probability density function; ( x2 + 2 y 2 ) cxy e , if x > 0, y > 0, f ( x, y ) = 0, otherwise. ( ) Evaluate the constant c and P X 2 > Y 2 . 20. Let PQ be a line segment of length and midpoint R. A point S is chosen at random on PQ. Let X , the distance from S to P, be a random variable having the uniform distribution on the interval ( 0, ) . Find the probability that PS , QS and PR form the sides of a triangle. 21. Let X 1 , X 2 , ... , X n be a random sample from a N ( , 1) distribution. For testing H 0 : = 10 against H1 : = 11, the most powerful critical region is X k , where X = 1 n n i =1 X i . Find k in terms of n such that the size of this test is 0.05. Further determine the minimum sample size n so that the power of this test is at least 0.95. 6 22. Consider the sequence {sn } , n 1, of positive real numbers satisfying the recurrence relation sn 1 + sn = 2 sn +1 for all n 2 . 1 | s2 s1 | for all n 1 . 2n 1 is a convergent sequence. (a) Show that | sn +1 sn | = (b) Prove that {sn } 23. The cumulative distribution function of a random variable 0, 1 1 + x3 , 5 F ( x) = 1 3 + ( x 1)2 , 5 1, ( X is given by if x < 0, ) if 0 x < 1, if 1 x < 2, if x 2. 3 1 Find P ( 0 < X < 2 ) , P ( 0 X 1) and P X . 2 2 ( ) 24. Let A and B be two events with P ( A | B ) = 0.3 and P A | B c = 0.4 . Find P( B | A) and P( B c | Ac ) in terms of P( B). If 1 1 1 9 and P( B c | Ac ) P( B | A) , then determine the value of P( B). 4 3 4 16 Optional Section B 25. Solve the initial value problem ( ) y y + y 2 x 2 + 2 x + 1 = 0, y (0) = 1. 26. Let y1 ( x) and y2 ( x) be the linearly independent solutions of x y + 2 y + x e x y = 0. If W ( x) = y1 ( x) y2 ( x) y2 ( x) y1 ( x) with W (1) = 2, find W (5). 1 27. (a) Evaluate (b) Evaluate W 1 x 0y 2 e x y dx dy. 9 Marks z dx dy dz , where W is the region bounded by the planes x = 0, y = 0, z = 0, z = 1 and the cylinder x 2 + y 2 = 1 with x 0, y 0. 6 Marks 28. A linear transformation T : is given by T ( x, y, z ) = ( 3 x + 11 y + 5 z , x + 8 y + 3 z ) . 3 2 Determine the matrix representation of this transformation relative to the ordered bases {(1, 0, 1) , ( 0, 1, 1) , (1, 0, 0 )} , {(1, 1) , (1, 0 )} . Also find the dimension of the null space of this transformation. 7 x2 + y2 , if x + y 0, 29. (a) Let f ( x, y ) = x + y 0, if x + y = 0. Determine if f is continuous at the point ( 0, 0 ) . 6 Marks (b) Find the minimum distance from the point (1, 2, 0 ) to the cone z = x + y . 2 2 2 9 Marks Optional Section C 30. Let X 1 , X 2 , ... , X n be a random sample from an exponential distribution with the probability density function; x 1 e , if x > 0, f ( x ; ) = 0, otherwise, where > 0. Derive the Cram r-Rao lower bound for the variance of any unbiased estimator of . 1n Hence, prove that T = X i is the uniformly minimum variance unbiased estimator of . n i =1 31. Let X 1 , X 2 , ... be a sequence of independently and identically distributed random variables with the probability density function; 1 2 x x e , if x > 0, f ( x) = 2 0, otherwise. 1 Show that lim P X 1 + ... + X n 3 n n . n 2 ( ( )) ( 32. Let X 1 , X 2 , ... , X n be a random sample from a N , 2 Find the value of b unknown. Tb = b n 1 n (X i =1 ) distribution, where both and 2 are that minimizes the mean squared error of the estimator 2 2 i X ) for estimating , where X = ( 1 n n X . i =1 i ) 33. Let X 1 , X 2 , ... , X 5 be a random sample from a N 2, 2 distribution, where 2 is unknown. Derive the most powerful test of size = 0.05 for testing H 0 : = 4 against H1 : 2 = 1. 2 34. Let X 1 , X 2 , ... , X n be a random sample from a continuous distribution with the probability density function; 2 x x2 , if x > 0, f (x; ) = e 0, otherwise, where > 0. Find the maximum likelihood estimator of and show that it is sufficient and an unbiased estimator of . 8

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