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IIT JAM 2021 : Mathematical Statistics (with Answer Keys)

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JAM 2021 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 1/21 JAM 2021 MATHEMATICAL STATISTICS - MS Special Instructions/ Useful Data det | Var , Poisson , , | | MLE ! max{ , , , min{ , , , ln MS } } The set of real numbers { , , , , = , , , } Determinant of a matrix Identity matrix of order , = , , First derivative of a real valued function Second derivative of a real valued function Complement of an event Probability of an event Conditional probability of an event given the occurrence of event The probability density/mass function of the random variable is Expectation of a random variable Variance of a random variable Continuous uniform distribution on the interval , , < < < Poisson distribution with mean , , Normal distribution with mean and variance , , , , Central chi-square distribution with degrees of freedom, = , , distribution with , degrees of freedom, , = , , Distribution function of , Absolute value of Maximum Likelihood Estimator , = , , , , and ! = ! , = , , , , and = , , ; = ! ! Maximum of real numbers , , , Minimum of real numbers , , , Natural logarithm of 2/21 JAM 2021 MATHEMATICAL STATISTICS - MS SECTION A MULTIPLE CHOICE QUESTIONS (MCQ) Q. 1 Q.10 carry one mark each. Q.1 The value of the limit lim ( + ) ( + ) is equal to (A) Q.2 (B) (C) Let : be a function defined by = + + Then, which of the following statements is TRUE? Q.3 (A) is both one-one and onto (B) is neither one-one nor onto (C) is one-one but NOT onto (D) is onto but NOT one-one (D) + , . The value of the limit lim is equal to (A) MS + (B) + cos (C) (D) 3/21 JAM 2021 Q.4 MATHEMATICAL STATISTICS - MS The value of the limit lim is equal to (A) Q.5 Let { = (C) (B) } (D) be a sequence of independent and identically distributed random variables with probability density function = { Then, the value of the limit lim ( is equal to Let be a , random variables (A) Q.7 Let ln = (B) (A) Q.6 , if < < . , otherwise (C) random variable and let and , then (B) be a = is equal to . If (C) + ) (D) is the correlation coefficient between the (D) 30 real matrix. Let ( ) , ( ) and ( ) be the eigenvectors of three distinct eigenvalues of , where corresponding to is a real number. Then, which of the following is NOT a possible value of ? (A) MS (B) (C) (D) 4/21 JAM 2021 Q.8 MATHEMATICAL STATISTICS - MS If the series = (A) (C) Q.9 = | converges absolutely, then which of the following series diverges? | (B) = (D) There are three urns, labeled, Urn , Urn balls, Urn contains + + = = and Urn . Urn ln white ball and 3 black balls and Urn + contains white balls and black contains white balls and black ball. Consider two coins with probability of obtaining head in their single trials as . and . . The two coins are tossed independently once, and an urn is selected according to the following scheme: Urn is selected if heads are obtained; Urn 3 is selected if tails are obtained; otherwise Urn is selected. A ball is then drawn at random from the selected urn. Then Urn is selected | the ball drawn is white is equal to Q.10 6 (B) 12 (C) 109 Let be a random variable with probability density function 109 = | | (A) (B) (C) (D) (D) 18 , < Then, which of the following statements is FALSE? MS 1 (A) 1 9 < . | | = | | | | sin | | sin = | | | | = = 5/21 JAM 2021 MATHEMATICAL STATISTICS - MS Q. 11 Q. 30 carry two marks each. Q.11 Let : be a function defined by , Let and , (C) (D) Q.12 Let { Then, , exists and is bounded at every , exists and exists and is bounded at every , Q.13 , = , , . , , , , lim (B) with respect to ) = and , , , be a sequence of independent and identically distributed is equal to (A) , is continuous at is NOT differentiable at } , . Then, which of the following statements is FALSE? , , (B) + , denote the first order partial derivatives of respectively, at the point (A) ={ , random variables. (C) (D) Consider a sequence of independent Bernoulli trials with probability of success in each trial as . The probability that three successes occur before four failures is equal to (A) MS 179 243 (B) 179 841 (C) 233 729 (D) 179 1215 6/21 JAM 2021 Q.14 MATHEMATICAL STATISTICS - MS Let and be independent following expectations is finite? (A) Q.15 (B) , = random variables and ( ) | | | | (C) . Then, which of the (D) Consider three coins having probabilities of obtaining head in a single trial as , and , respectively. A player selects one of these three coins at random (each coin is equally likely to be selected). If the player tosses the selected coin five times independently, then the probability of obtaining two tails in five tosses is equal to (A) (C) Q.16 Let 85 384 125 (D) 384 255 384 64 384 be a random variable having the probability density function Define MS (B) ={ , , > . = [ ], where [ ] denotes the largest integer not exceeding . Then, (A) + (B) (C) + (D) is equal to + + 7/21 JAM 2021 Q.17 MATHEMATICAL STATISTICS - MS Let Let be a continuous random variable having the moment generating function = Then, the value of (A) Q.18 Let let 10 > ln , , and , + = ln is equal to 19 (B) 3 , + = (C) 3 . ln 13 = is (B) is (C) does NOT exist (D) is , , unknown. Let (A) (B) + + 17 3 3 . , is unknown and be a random sample from = max{ statements is TRUE? MS (A) , . (D) Then, the uniformly minimum variance unbiased estimator of Let > be a random sample from Poisson , where = Q.19 , , , } and = min{ , , + , , , where , is }. Then, which of the following is the unique MLE of is an MLE of (C) MLE of (D) + does NOT exist is an MLE of 8/21 JAM 2021 Q.20 MATHEMATICAL STATISTICS - MS Let and be random variables having chi-square distributions with and degrees of freedom, respectively. Then, which of the following statements is TRUE? (A) (B) (C) (D) Q.21 Let Let (A) Q.22 Let > . > . < > > < < > > . > < , be a random vector with joint moment generating function = + . Then, Var , = + MS is equal to , < (B) < , < (C) < min{ , }. (D) be a continuous random variable with distribution function for some real constant . Then, (A) > . (B) ={ , , is equal to , if (C) if if < < , (D) 9/21 JAM 2021 Q.23 MATHEMATICAL STATISTICS - MS Let , , , function where test of size be a random sample from an exponential distribution with probability density ; = { , is unknown. Let for testing : = against , where for any critical region Q.24 , , , , be the power of the most powerful : = . > = > corresponds to the : = against : = against (A) most powerful test of size for testing (B) most powerful test of size for testing (C) most powerful test of size (D) most powerful test of size for testing for testing }, is a fixed point such that = . Then, the : = : = against : = against : = : = : = Let = = ( ) Then, which of the following statements is TRUE? (A) Let , | Then, (A) = (C) MS > , otherwise , , Q.25 , be fixed and let Consider the critical region ={ 1 2 , and and = = (B) = (D) ( ) . = = be four events such that = , = , , ; ( | is equal to (B) 2 3 | | = , , = , , ; and (C) 5 6 (D) = 7 12 10/21 JAM 2021 Q.26 MATHEMATICAL STATISTICS - MS Let = and define recursively = + Then, which of the following statements is TRUE? (A) (B) (C) (D) Q.27 { { { { } is monotone increasing, and lim } is monotone decreasing, and lim } is decreasing, and lim Consider the problem of testing = { : = = . = } is non-monotone, and lim , = : against , and , otherwise = { based on a sample of size , where , , Then, the probability of Type II error of the most powerful test of size (A) Q.28 For 0.81 (B) 0.91 (C) , consider the system of linear equations + + + + + 0.1 otherwise = . is equal to (D) 1 = + = = in the unknowns , and . Then, which of the following statements is TRUE? MS = (A) The given system has a unique solution for (B) The given system has infinitely many solutions for (C) The given system has a unique solution for (D) The given system has infinitely many solutions for = = = 11/21 JAM 2021 Q.29 MATHEMATICAL STATISTICS - MS Let { } following conditions imply the divergence of { (A) (B) (C) (D) Q.30 Let , { } = lim { and ? . Then, which of the is non-increasing converges, where + } } = = = and + , for all > converges be three events such that = , Then, which of the following statements is FALSE? MS , for all be a sequence of real numbers such that (A) (B) (C) (D) = and = . 12/21 JAM 2021 MATHEMATICAL STATISTICS - MS SECTION - B MULTIPLE SELECT QUESTIONS (MSQ) Q. 31 Q. 40 carry two marks each. Q.31 Consider the linear system and is an system = = , where is an matrix, vector. Further, suppose there exists an is an vector vector of unknowns such that the linear has NO solution. Then, which of the following statements is/are necessarily TRUE? If and (B) If (C) , then Rank < Rank < (D) Q.32 is the first column of , then the linear system = (A) If Let be a > , then the linear system = real matrix such that has a solution other than has a unique solution = and the sum of the entries in each row of is . Then, which of the following statements is/are necessarily TRUE? MS (A) (B) The set { (C) The characteristic polynomial, (D) cannot be an orthogonal matrix is an invertible matrix : = } has at least two elements ( is a column vector) , of + + has as a factor 13/21 JAM 2021 Q.33 MATHEMATICAL STATISTICS - MS Let , , , be a random sample from : problem of testing against < likelihood ratio test of size statements is/are TRUE? < (A) > , for all > (B) < , for all > (C) : > . Let { , for testing , , , , , . Then, which of the following against > = , : > is = The critical region of the likelihood test of size where is a fixed point such that > }, , . is = = , < }, , . Consider the function If MS denote the power function of the : is a fixed point such that { Q.34 , where , is unknown. Consider the The critical region of the likelihood test of size where (D) , ={ , + , = + + , , . = }, then which of the following statements is/are TRUE? (A) The maximum value of on is (B) The minimum value of on is (C) The maximum value of on is (D) The minimum value of on is + + 14/21 JAM 2021 Q.35 MATHEMATICAL STATISTICS - MS Let : be a twice differentiable function. Then, which of the following statements is/are necessarily TRUE? is continuous (A) (B) (D) Let , = is bounded on , , , then is bounded on [ , (C) Q.36 If ] , = , has a solution in be independent and identically distributed random variables with probability density function ={ , , otherwise . Then, which of the following random variables has/have finite expectation? (A) Q.37 A sample of size which are red balls and is/are TRUE? (A) + (C) = = (D) min{ is drawn randomly (without replacement) from an urn containing selected sample. If = lim MS (C) (B) are black balls. Let and = lim Var (B) (D) } , , balls, of denote the number of red balls in the , then which of the following statements = = 15/21 JAM 2021 Q.38 MATHEMATICAL STATISTICS - MS Let , , , where be a random sample from a distribution with probability density function , is unknown. If ; = { , = min{ of the following statements is/are TRUE? , (A) max{| (C) (D) Q.39 Let , If , , = = (A) MS |, | |, , | } and , = max{ , , , }, then which |} is a complete and sufficient statistic for does NOT depend on Distribution of where , , | |> is jointly sufficient for is an MLE of (B) , , be a random sample from a distribution with probability density function , is unknown. ; = { , / , > , otherwise , then which of the following statements is/are TRUE? is the unique uniformly minimum variance unbiased estimator of (B) is the unique uniformly minimum variance unbiased estimator of (C) (D) is the MLE of = is the unique uniformly minimum variance unbiased estimator of 16/21 JAM 2021 Q.40 MATHEMATICAL STATISTICS - MS Let , , , where be a random sample from a distribution with probability density function ; = { , otherwise , , is unknown. Then, which of the following statements is/are TRUE? (A) Cramer-Rao lower bound, based on , , , , for the estimand is 9 (B) Cramer-Rao lower bound, based on , , , , for the estimand is (C) There does NOT exist any unbiased estimator of bound (D) MS , There exists an unbiased estimator of which attains the Cramer-Rao lower which attains the Cramer-Rao lower bound 17/21 JAM 2021 MATHEMATICAL STATISTICS - MS SECTION C NUMERICAL ANSWER TYPE (NAT) Q. 41 Q. 50 carry one mark each. Q.41 Let , and + Q.42 be a ___________________. Q.43 Let ={ , Let : , then the value of and > , then the value of is }. Let be the value of the integral . is equal to ____________________________. and be the probability mass functions given by . . . . . sample . If { . . Consider the problem of testing the null hypothesis MS = Then, Q.45 ]. If }. Then, the value of the integral Let = { , , = det matrix. If is _________________________. Q.44 =[ is _______________________. = Let be the eigenvalues of . . : . . against . : based on a single and , respectively, denote the size and power of the test with critical region > }, then + is equal to _________________. 18/21 JAM 2021 Q.46 MATHEMATICAL STATISTICS - MS Let , , , , be an observed random sample of size from a distribution with probability density function , ; = { , , otherwise ] is unknown. Then, the maximum likelihood estimate of based on the observed sample is equal to ____________________. Q.47 Let Then, Q.48 Let Then, Q.49 Let Then, Q.50 Let = lim + + is equal to ____________________________. be a random variable having the probability density function = + is equal to __________________. , < < . be a random variable with moment generating function + = + + is equal to ________________________. denote the length of the curve Then, the value of MS = = ln sec from + = is equal to ______________. to , . = . 19/21 JAM 2021 MATHEMATICAL STATISTICS - MS Q.51 Q. 60 carry two marks each. Q.51 be the region bounded by the parallelogram with vertices at the points Let , Q.52 and , . Then, the value of the integral Let = { , : of , be < < + ={ , , , Let Let then Q.54 and = sgn = and , and is equal to ________________. random variables. Define , if = { , if , if sgn sgn < = > . If the correlation coefficient between and is , Let Let lim , , = and = , = , , . is equal to _______________________. be four independent events such that Let the random vector Let MS , , otherwise = , = . Let be the probability that at most two events among Then, is equal to _____________. Q.56 , is equal to _____________. is equal to __________________________. Then, Q.55 be independent , } and let the joint probability density function Then, the covariance between the random variables Q.53 + , = , + = { . If , , , = , , , ; = , and = and occur. have the joint probability mass function ( )( ) = and + , ( ) = Var + , , then + = , , , otherwise is equal to ___________. 20/21 JAM 2021 Q.57 MATHEMATICAL STATISTICS - MS Let If Q.58 = { , : , min{sin , cos } is the area of , then the value of Q.60 is equal to __________________. The number of real roots of the polynomial = is ________________. Q.59 max{sin , cos }}. Let = lim Let : , If = lim + sin . Then, ln be defined by , then + is equal to ___________________. = + . is equal to ___________________. END OF THE QUESTION PAPER MS 21/21 Answer Key of JAM-2021 Mathematical Statistics (MS) Paper Note: Question numbers pertain to the question paper published on the JAM 2021 website Q. No. Answer Q. No. Answer 1 D 31 C 2 A 32 B, C 3 D 33 A 4 B 34 C, D 5 B 35 B, C 6 B 36 B, C, D 7 C 37 A, D 8 D 38 A, C, D 9 A 39 A, D 10 D 40 A, D 11 C 41 7 12 B 42 2500 13 C 43 2 14 A 44 8 15 A 45 13 16 B 46 3 17 B 47 10 18 B 48 147 19 Marks to All 49 2 20 A 50 6 21 C 51 42 22 A 52 1 23 D 53 2 24 A 54 3 25 C 55 218 26 B 56 225 27 A 57 8 28 C 58 3 29 C 59 6 30 C 60 21

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