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IIT JAM 2019 : Mathematical Statistics

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JAM 2019 MATHEMATICAL STATISTICS - MS Paper Specific Instructions 1. The examination is of 3 hours duration. There are a total of 60 questions carrying 100 marks. The entire paper is divided into three sections, A, B and C. All sections are compulsory. Questions in each section are of different types. 2. Section A contains a total of 30 Multiple Choice Questions (MCQ). Each MCQ type question has four choices out of which only one choice is the correct answer. Questions Q.1 Q.30 belong to this section and carry a total of 50 marks. Q.1 Q.10 carry 1 mark each and Questions Q.11 Q.30 carry 2 marks each. 3. Section B contains a total of 10 Multiple Select Questions (MSQ). Each MSQ type question is similar to MCQ but with a difference that there may be one or more than one choice(s) that are correct out of the four given choices. The candidate gets full credit if he/she selects all the correct answers only and no wrong answers. Questions Q.31 Q.40 belong to this section and carry 2 marks each with a total of 20 marks. 4. Section C contains a total of 20 Numerical Answer Type (NAT) questions. For these NAT type questions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor. No choices will be shown for these type of questions. Questions Q.41 Q.60 belong to this section and carry a total of 30 marks. Q.41 Q.50 carry 1 mark each and Questions Q.51 Q.60 carry 2 marks each. 5. In all sections, questions not attempted will result in zero mark. In Section A (MCQ), wrong answer will result in NEGATIVE marks. For all 1 mark questions, 1/3 marks will be deducted for each wrong answer. For all 2 marks questions, 2/3 marks will be deducted for each wrong answer. In Section B (MSQ), there is NO NEGATIVE and NO PARTIAL marking provisions. There is NO NEGATIVE marking in Section C (NAT) as well. 6. Only Virtual Scientific Calculator is allowed. Charts, graph sheets, tables, cellular phone or other electronic gadgets are NOT allowed in the examination hall. 7. The Scribble Pad will be provided for rough work. MS 1/15 JAM 2019 MATHEMATICAL STATISTICS - MS Special Instructions / Useful Data The set of all real numbers Transpose of the matrix 1 { [ ] , = 1, 2, , } Derivative of the differentiable function identity matrix Probability of the event Expectation of the random variable Variance of the random variable Independently and identically distributed Continuous uniform distribution on ( , ), < < < Exponential distribution with probability density function, for > 0, ( ) , > 0, ( ) = { 0, otherwise 2 ) Normal distribution with mean and variance 2 ( , ( ) ( ) ( ) i.i.d. ( , ) ( ) 2 , ! 1 2 2 2 Central Chi-squared distribution with degrees of freedom A constant such that ( > , ) = , where has Student s -distribution with degrees of freedom ( 1) 3 2 1 for = 1, 2, 3 , and 0! = 1 (1.65) = 0.950, (1.96) = 0.975 4, 0.05 = 2.132, 4, 0.10 = 1.533 MS 2/15 JAM 2019 MATHEMATICAL STATISTICS - MS SECTION A MULTIPLE CHOICE QUESTIONS (MCQ) Q. 1 Q.10 carry one mark each. Q.1 Let { } 1 be a sequence of positive real numbers. Which one of the following statements is always TRUE? (A) If { } 1 is a convergent sequence, then { } 1 is monotone (B) If { 2 } 1 is a convergent sequence, then the sequence { } 1 does not converge (C) If the sequence {| +1 |} 1 converges to 0, then the series =1 is convergent (D) If { } 1 is a convergent sequence, then { } 1 is also a convergent sequence Q.2 Consider the function ( , ) = 3 3 2 , , . Which one of the following statements is TRUE? (A) (B) (C) (D) Q.3 has a local minimum at (0, 0) has a local maximum at (0, 0) has global maximum at (0, 0) has a saddle point at (0, 0) 4 If ( ) = 3 4 + 2 , for , then (1) equals (A) 3 5 Q.4 (D) 3 5 2 3 (B) 4 (C) 3 5 (D) 3 7 3 2 3 Two biased coins 1 and 2 have probabilities of getting heads and , respectively, 3 4 when tossed. If both coins are tossed independently two times each, then the probability of getting exactly two heads out of these four tosses is (A) MS (C) 2 5 1 1 2 0 Let : 2 2 be a linear transformation such that ([ ]) = [ ] and ([ ]) = [ ]. 0 2 1 1 3 1 2 3 Suppose that [ ] = [ ] + [ ] and ([ ]) = [ ]. Then + + + equals 2 2 1 2 (A) Q.5 (B) 2 5 1 4 (B) 37 144 (C) 41 144 (D) 49 144 3/15 JAM 2019 Q.6 MATHEMATICAL STATISTICS - MS Let be a discrete random variable with the probability mass function 2 , ( = ) = , , { 0, = 1, 2, = 0, = 1, 2, otherwise, where and are positive real numbers. If (| | 1) = 3/4, then ( ) equals (A) Q.7 Q.8 (B) 12 1 (C) 6 1 (D) 3 1 2 Let be a Poisson random variable and ( = 1) + 2 ( = 0) = 12 ( = 2). Which one of the following statements is TRUE? (A) 0.40 < ( = 0) 0.45 (B) 0.45 < ( = 0) 0.50 (C) 0.50 < ( = 0) 0.55 (D) 0.55 < ( = 0) 0.60 Let 1 , 2 , be a sequence of i.i.d. discrete random variables with the probability mass function (log e 2) , = 0,1,2, , ( 1 = ) = { 2( !) 0, otherwise. If = 1 + 2 + + , then which one of the following sequences of random variables converges to 0 in probability? (A) Q.9 1 (B) log 2 log 2 (C) log 2 (D) log 2 Let 1 , 2 , , be a random sample from a continuous distribution with the probability density function ( ) = 1 1 2 2 1 2 2 [ 2( 2 ) + 2( 4 ) ] , < < . If = 1 + 2 + + , then which one of the following is an unbiased estimator of ? (A) MS (B) 2 (C) 3 (D) 4 4/15 JAM 2019 MATHEMATICAL STATISTICS - MS Q.10 Let 1 , 2 , , be a random sample from a ( , 1) distribution. Instead of observing 1 , 2 , , , we observe 1 , 2 , , , where = , = 1, 2, , . To test the hypothesis 0 : = 1 against 1 : 1 based on the random sample 1 , 2 , , , the rejection region of the likelihood ratio test is of the form, for some 1 < 2 , (A) =1 1 or =1 2 (B) 1 =1 2 (C) 1 =1 log 2 (D) =1 log 1 or =1 log 2 Q. 11 Q. 30 carry two marks each. Q.11 =4 (A) Q.12 6 2 4 +3 equals 5 (B) 3 2 1 1 (B) 2 (D) 9 2 equals ( + )1/ log (A) 7 1 1+2+ + lim (C) 1 (C) (D) Q.13 A possible value of for which the equation 4 + 3 + 1 = 0 has no real root is (A) Q.14 11 5 (B) 3 (C) 2 2 Let the Taylor polynomial of degree 20 for 1 (1 )3 (D) 5 2 at = 0 be 20 =0 . Then 15 is (A) 136 (B) 120 (C) 60 (D) 272 Q.15 The length of the curve = 3 4/3 3 2/3 + 7 from = 1 to = 8 equals 4 (A) MS 99 8 (B) 117 8 8 (C) 99 4 (D) 117 4 5/15 JAM 2019 MATHEMATICAL STATISTICS - MS Q.16 The volume of the solid generated by revolving the region bounded by the parabola = 2 2 + 4 and the line = 6 about the line = 6 is (A) 78 15 (B) 91 15 (C) 64 15 (D) 117 15 Q.17 Let be a 3 3 non-null real matrix. If there exist a 3 2 real matrix and a 2 3 real matrix such that = , then (A) = has a unique solution, where 3 (B) there exists 3 such that = has no solution (C) there exists a non-zero 3 such that = has a unique solution (D) there exists a non-zero 3 such that = has a unique solution Q.18 1 If = [0 2 (A) (3, 2) 0 1 2 1] and 6 1 = 3 + 2 , then the ordered pair ( , ) is 0 1 (B) (2, 3) (C) (4, 5) (D) (5, 4) Q.19 Let , and be any three events with ( ) = 0.3, ( | ) = 0.2, ( | ) = 0.1 and ( | ) = 0.05. Then ( ( )) equals (A) 0.155 (B) 0.175 (C) 0.225 (D) 0.255 Q.20 Let and be any two independent events with 0 < ( ) < 1 and 0 < ( ) < 1. Which one of the following statements is NOT TRUE? (A) (B) (C) (D) (Neither nor occurs) = ( ( ) 1)( ( ) 1) (Exactly one of and occurs) = ( ) + ( ) ( ) ( ) ( occurs but does not occur) = ( ) ( ) ( occurs given that does not occur) = ( ) Q.21 Let be a continuous random variable with the probability density function 1 7 2 ( ) = {3 , > 0, 0, otherwise. Then the distribution of the random variable = 2 2 is (A) 22 MS (B) 42 (C) 62 (D) 82 6/15 JAM 2019 MATHEMATICAL STATISTICS - MS Q.22 Let be a continuous random variable with the probability density function ( ) = , < < . (1 + )2 Then ( ) and ( > 1), respectively, are (A) 1 and (1 + ) 1 (C) 2 and (2 + 2 ) 1 (B) 0 and 2(1 + ) 2 (D) 0 and (1 + ) 1 Q.23 The lifetime (in years) of bulbs is distributed as an (1) random variable. Using Poisson approximation to the binomial distribution, the probability (round off to 2 decimal places) that out of the fifty randomly chosen bulbs at most one fails within one month equals (A) 0.05 (B) 0.07 (C) 0.09 (D) 0.11 Q.24 Let follow a beta distribution with parameters (> 0) and 2. If ( 1) = 1, then 2 2 ( ) equals (A) 1 10 (B) 1 20 (C) 1 (D) 25 1 40 Q.25 Let 1 , 2 and 3 be i.i.d. (0,1) random variables. Then ( 1 > 2 + 3 ) equals (A) 1 6 (B) 1 4 (C) 1 (D) 3 1 2 Q.26 Let and be i.i.d. (0, 1) random variables. Then ( | > ) equals (A) 1 3 (B) 1 2 (C) 2 (D) 3 3 4 Q.27 Let 1 and 1 be the observed values of a random sample of size two from ( , ) distribution. The maximum likelihood estimate of is (A) 0 MS (B) 2 (C) 5 1 2 (D) 5 1 2 7/15 JAM 2019 MATHEMATICAL STATISTICS - MS Q.28 Let 1 and 2 be a random sample from a continuous distribution with the probability density function 1 , > , ( ) = { 0, otherwise, where > 0. If (1) = min{ 1 , 2 } and = statements is TRUE? (A) (B) (C) (D) ( 1 + 2 ) 2 , then which one of the following ( , (1) ) is sufficient and complete ( , (1) ) is sufficient but not complete ( , (1) ) is complete but not sufficient ( , (1) ) is neither sufficient nor complete Q.29 Let 1 , 2 , , be a random sample from a continuous distribution with the probability density function ( ). To test the hypothesis 0 : ( ) = 1 2 , < < against 1 : ( ) = 2| | , < < , the rejection region of the most powerful size test is of the form, for some > 0, (A) =1( 1)2 (B) =1( 1)2 (C) =1(| | 1)2 (D) =1(| | 1)2 Q.30 Let 1 , 2 , , be a random sample from a ( , 1) distribution. To test 0 : = 0 1 3 against 1 : = 1, assume that the critical region is given by =1 > . Then the 4 minimum sample size required so that (Type I error) 0.05 is (A) 3 MS (B) 4 (C) 5 (D) 6 8/15 JAM 2019 MATHEMATICAL STATISTICS - MS SECTION - B MULTIPLE SELECT QUESTIONS (MSQ) Q. 31 Q. 40 carry two marks each. Q.31 Let { } 1 be a sequence of positive real numbers such that the series =1 converges. Which of the following statements is (are) always TRUE? (A) The series =1 +1 converges (B) lim = 0 2 (C) The series =1 sin converges (D) The series =1 1+ converges Q.32 Let : be continuous on and differentiable on ( , 0) (0, ). Which of the following statements is (are) always TRUE? (A) If is differentiable at 0 and (0) = 0, then has a local maximum or a local minimum at 0 (B) If has a local minimum at 0, then is differentiable at 0 and (0) = 0 (C) If ( ) < 0 for all < 0 and ( ) > 0 for all > 0, then has a global maximum at 0 (D) If ( ) > 0 for all < 0 and ( ) < 0 for all > 0, then has a global maximum at 0 Q.33 Let be a 2 2 real matrix such that every non-zero vector in 2 is an eigenvector of . 2 Suppose that 1 and 2 denote the eigenvalues of and [ 2] = [ ] for some 3 . Which of the following statements is (are) TRUE? (A) 1 2 (B) 1 2 = 2 (C) 2 is an eigenvalue of (D) 3 is an eigenvalue of Q.34 Let be an non-null real skew-symmetric matrix, where is even. Which of the following statements is (are) always TRUE? (A) (B) (C) (D) MS = has infinitely many solutions, where = has a unique solution for every non-zero If = ( + )( ) 1, then = The sum of all the eigenvalues of is zero 9/15 JAM 2019 MATHEMATICAL STATISTICS - MS Q.35 Let be a random variable with the cumulative distribution function 0, 1 + 2 , 10 ( ) = 3 + 2 , 10 { 1, < 0, 0 < 1, 1 < 2, 2. Which of the following statements is (are) TRUE? (A) (1 < < 2) = (C) (1 < 2) = 3 10 1 2 (B) (1 < 2) = 3 (D) (1 2) = 4 5 5 Q.36 Let and be i.i.d. ( ) random variables. If = max{ , 0}, then which of the following statements is (are) TRUE? (A) ( = 0) = 1 2 (B) The cumulative distribution function of is ( ) = { 0, 1 1 , 2 < 0, 0 (C) ( = 0) = 0 (D) The cumulative distribution function of is ( ) = { 0, < 0, /2 1 , 0 Q.37 Let the discrete random variables and have the joint probability mass function 2 ( = , = ) = { ! ! , = 0,1,2, ; = 0,1,2, , 0, otherwise. Which of the following statements is (are) TRUE? (A) The marginal distribution of is Poisson with mean 2 (B) The random variables and are independent (C) The covariance between and + 3 is 1 (D) ( = ) = ( + 1) ( = + 1) for = 0,1,2, MS 10/15 JAM 2019 MATHEMATICAL STATISTICS - MS Q.38 Let 1 , 2 , be a sequence of i.i.d. continuous random variables with the probability density function 1 1 2( ) 2 , 2 , ( ) = { 2 0, otherwise. If = 1 + 2 + + and = / , then the distributions of which of the following sequences of random variables converge(s) to a normal distribution with mean 0 and a finite variance? (A) (B) (C) ( 1) (D) ( 1) 2 Q.39 Let 1 , 2 , , be a random sample from a ( , 0) distribution, where < 0. If = min{ 1 , 2 , , }, then which of the following sequences of estimators is (are) consistent for ? (A) (B) 1 (C) + 1 (D) 1 1 2 Q.40 Let 1 , 2 , , be a random sample from a continuous distribution with the probability density function, for > 0, 2 ( ) = {2 0, 1 , > 0, otherwise. 3 To test the hypothesis 0 : = 2 against 1 : = 4 at the level (0 < < 1), which of the following statements is (are) TRUE? (A) The most powerful test exists for each value of (B) The most powerful test does not exist for some values of (C) If the most powerful test exists, it is of the form: Reject 0 if 12 + 22 + + 2 for some > 0 (D) If the most powerful test exists, it is of the form: Reject 0 if 12 + 22 + + 2 for some > 0 MS 11/15 JAM 2019 MATHEMATICAL STATISTICS - MS SECTION C NUMERICAL ANSWER TYPE (NAT) Q. 41 Q. 50 carry one mark each. Q.41 +1+ +2+ + + (round off to 2 decimal places) equals __________ lim Q.42 Let : [0, 2] be such that | ( ) ( )| | |4/3 for all , [0, 2]. 2 If 0 ( ) = 2 1 , then 2019 =1 ( ) equals __________ 3 Q.43 The value (round off to 2 decimal places) of the double integral 9 3 0 1 1 + 3 equals __________ Q.44 5 If 3 2 3 [ 2 3 5 3 is a real orthogonal matrix, then 2 + 2 + 2 + 2 equals ________ 1] Q.45 Two fair dice are tossed independently and it is found that one face is odd and the other one is even. Then the probability (round off to 2 decimal places) that the sum is less than 6 equals __________ Q.46 Let be a random variable with the moment generating function ( ) = ( 2 + 2 2 2 ) , < < . 2 Using Chebyshev s inequality, the upper bound for (| | > ) equals __________ 3 MS 12/15 JAM 2019 MATHEMATICAL STATISTICS - MS Q.47 In a production line of a factory, each packet contains four items. Past record shows that 20% of the produced items are defective. A quality manager inspects each item in a packet and approves the packet for shipment if at most one item in the packet is found to be defective. Then the probability (round off to 2 decimal places) that out of the three randomly inspected packets at least two are approved for shipment equals __________ Q.48 Let be the number of heads obtained in a sequence of 10 independent tosses of a fair coin. The fair coin is tossed again number of times independently, and let be the number of heads obtained in these number of tosses. Then ( + 2 ) equals __________ Q.49 Let 0, 1, 0, 0, 1 be the observed values of a random sample of size five from a discrete distribution with the probability mass function ( = 1) = 1 ( = 0) = 1 , where > 0. The method of moments estimate (round off to 2 decimal places) of equals __________ Q.50 Let 1 , 2 , 3 be a random sample from ( 1 , 2 ) distribution and 1 , 2 , 3 be a random sample from ( 2 , 2 ) distribution. Also, assume that ( 1 , 2 , 3 ) and ( 1 , 2 , 3 ) are independent. Let the observed values of 2 1 2 1 3 =1 [ ( 1 + 2 + 3 )] 3 and 3 =1 [ ( 1 + 2 + 3 )] be 1 and 5, respectively. The length (round off to 2 decimal 3 places) of the shortest 90% confidence interval of 1 2 equals __________ Q. 51 Q. 60 carry two marks each. Q.51 1 lim [ (1 + ) ] equals __________ Q.52 For any real number , let [ ] be the greatest integer less than or equal to and let { } = [ ]. For = 1, 2, , and for , let [ 2 ( ) = { sin 1, ], 0, = 0, and { 2 1 ( ) = { sin 1, }, 0, = 0. Then lim 100 =1 ( ) equals __________ 0 MS 13/15 JAM 2019 MATHEMATICAL STATISTICS - MS Q.53 The volume (round off to 2 decimal places) of the region in the first octant ( 0, 0, 0 ) bounded by the cylinder 2 + 2 = 4 and the planes = 2 and + = 4 equals __________ Q.54 0 0 2 If = 2 and = 1, then the determinant of [3 10 ] 0 0 2 7 2 equals _________ Q.55 In an ethnic group, 30% of the adult male population is known to have heart disease. A test indicates high cholesterol level in 80% of adult males with heart disease. But the test also indicates high cholesterol levels in 10% of the adult males with no heart disease. Then the probability (round off to 2 decimal places), that a randomly selected adult male from this population does not have heart disease given that the test indicates high cholesterol level, equals __________ Q.56 Let be a continuous random variable with the probability density function 2 , 0 < < 1, ( ) = { 4 , 1, 0, otherwise, where and are positive real numbers. If ( ) = 1, then ( 2 ) equals __________ Q.57 Let and be jointly distributed continuous random variables, where is positive valued with ( 2 ) = 6. If the conditional distribution of given = is (1 , 1 + ), then ( ) equals __________ 2 Q.58 Let 1 , 2 , , 10 be i.i.d. (0, 1) random variables. If = 12 + 22 + + 10 , then E( 1 ) equals __________ Q.59 Let 1 , 2 , 3 be a random sample from a continuous distribution with the probability density function ( ) , > , ( ) = { 0, otherwise. Let (1) = min{ 1 , 2 , 3 } and > 0 be a real number. Then ( (1) , (1) ) is a 97% confidence interval for , if (round off to 2-decimal places) equals __________ MS 14/15 JAM 2019 MATHEMATICAL STATISTICS - MS Q.60 Let 1 , 2 , 3 , 4 be a random sample from a discrete distribution with the probability mass function ( = 0) = 1 ( = 1) = 1 , for 0 < < 1. To test the hypothesis 3 4 0 : = against 1 : = , 4 5 consider the test: Reject 0 if 1 + 2 + 3 + 4 > 3. Let the size and power of the test be denoted by and , respectively. Then + (round off to 2 decimal places) equals __________ END OF THE QUESTION PAPER MS 15/15

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