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IIT JAM 2018 : Mathematical Statistics (with Answer Keys)

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JAM 2018 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 1/17 JAM 2018 Mathematical Statistics - MS Special Instructions/Useful Data All angles are in radian Set of all real numbers { , , , : } , Transpose of the matrix Derivative of the function Probability of the event Expectation of the random variable Variance of the random variable i.i.d. , ! , Continuous uniform distribution on The gamma function MS Independently and identically distributed = The factorial function != , < < , > / < 2/17 JAM 2018 Mathematical Statistics - MS SECTION A MULTIPLE CHOICE QUESTIONS (MCQ) Q. 1 Q.10 carry one mark each. Q.1 } Let { + Then (A) . Q.2 (B) there exists a natural number such that (C) there exists a natural number such that (D) there exists a natural number such that , and, for + . + = , for all natural number > < . = + The value of lim ( + ) is (A) Q.3 = be a sequence of real numbers such that Let { } = max{ (A) neither { (B) { (C) { (B) } } (D) both { and { , } } (C) = min{ } nor { } converges but { , and { } }. Then does not converge } converge , define converges does not converge but { } (D) be two convergent sequences of real numbers. For } and converges Q.4 Let = [ (A) (B) (C) (D) MS ]. If is the + = + = + zero matrix, then = + identity matrix and is the = 3/17 JAM 2018 Q.5 Mathematical Statistics - MS Let be a random variable with the probability density function = If Q.6 and = = { , , is , be a random variable with the distribution function , Then = equals (A) Let (A) Let + = . = and , be i.i.d. (B) , = , = , (C) (D) , , < , + = + . (D) random variables. Then + + + (C) = and = , < , (C) (B) , > , > , otherwise. Let = { + Q.8 , , , Q.7 (A) (B) , , then equals (D) be the observed values of a random sample of size from a discrete distribution with the probability mass function ; = = = , { , = , , = , = , , where [ , ] is the unknown parameter. Then the maximum likelihood estimate of is (A) MS (B) (C) (D) 4/17 JAM 2018 Q.9 Mathematical Statistics - MS Consider four coins labelled as , , and . Suppose that the probability of obtaining a head in a single toss of the coin is , = , , , . A coin is chosen uniformly at random and flipped. Given that the flip resulted in a head , the conditional probability that the coin was labelled either or equals (A) Q.10 (B) (C) = Consider the linear regression model standard normal random variables. Given that = = . , ( = = )( ( = = (A) and (B) and (C) and (D) and and = ) = . and )= . , , respectively, are the maximum likelihood estimates of MS = + ; = , , , , where s are i.i.d. + = . , (D) 5/17 JAM 2018 Mathematical Statistics - MS Q. 11 Q. 30 carry two marks each. Q.11 Let : [ , ] be defined by than or equal to . Then (A) Q.12 (C) (D) is continuous everywhere except at Let , (A) is discontinuous at , , , be defined by (B) (C) (D) Q.13 = , for all = c s = and si . Then in = for exactly two values of ={ , Then the minimum value of on (A) : )= equals } and the function : + (B) Let = [ (C) satisfies (D) satisfies , (C) ] be an orthogonal matrix with , , (A) is a skew-symmetric matrix MS for exactly one value of Consider the domain (B) is the = for more than two values of ( , Q.14 where [ ] denotes the greatest integer less is continuous at , , is discontinuous at , , (B) + [si ] , +| | = = + , defined by . (D) as its column vectors. Then identity matrix = = 6/17 JAM 2018 Q.15 Mathematical Statistics - MS Let : [ , ] be defined by Now, define : [ , ] by Then Q.16 = = (A) is differentiable at (B) is differentiable at (C) is not differentiable at (D) is differentiable at = { = = and and and = and If , and are real numbers such that of + + , < , < . , + , = , for < . = = = + + = and + = , then the value (A) cannot be computed from the given information (B) equals (C) equals (D) equals Q.17 Let = [ then ={ , , : [ ] = [ ]} and (A) the dimension of equals (B) the dimension of equals (C) the dimension of (D) MS ]. If ={ , , : [ ] = [ ]} , equals ={ , , } 7/17 JAM 2018 Q.18 Mathematical Statistics - MS Let be a non-zero, skew-symmetric real matrix. If is the identity matrix, then (A) is invertible (B) the matrix + is invertible (C) there exists a non-zero real number such that (D) all the eigenvalues of are real Q.19 Let be a random variable with the moment generating function (A) 0 Let (B) / , . (C) (D) be a discrete random variable with the moment generating function Then = (A) (C) Q.21 = , where is the set of rational numbers, equals Then Q.20 + is not invertible } Let { + = = + (B) (D) , . = = = be a sequence of independent random variables with having the probability density function as = { Then lim [ ( equals (A) MS + (B) , / > )+ (C) > , > , otherwise. + ] (D) 8/17 JAM 2018 Q.22 Mathematical Statistics - MS Let (A) Q.23 Let Let (A) Q.25 and | = , , Let and (C) MS > equals = and , = { | = , , otherwise. . Then , (B) is , , , have the joint probability mass function = = | = and = , equals = = { , + , = , , ; = , , otherwise. (C) (B) (D) be two independent standard normal random variables. Then the probability density | | function of (A) + (D) (A) (D) (D) ! equals have the joint probability density function = (C) Let (B) Then Q.26 (C) be a standard normal random variable. Then (C) Let (B) (A) Q.24 + be a Poisson random variable with mean . Then = | | is / { = { , , , , > , otherwise > , otherwise (B) = { (D) = { , , + / , , > , otherwise > , otherwise 9/17 JAM 2018 Q.27 Mathematical Statistics - MS Let and have the joint probability density function , Then the correlation coefficient between (B) (A) Q.28 = , Let = = { , and , < < < , otherwise. equals (C) (D) = be the observed values of a random sample of size three from a and discrete distribution with the probability mass function ; = { , + , , , , }, = { + , , = otherwise, where = { , , } is the unknown parameter. Then the method of moment estimate of is (A) Q.29 Let (B) (C) be a random sample from a discrete distribution with the probability mass function where = { , (A) ; = = { , , = : = against (C) (D) and otherwise, : = = . . Then the uniformly most powerful test rejects (B) Let = , , , , } is the unknown parameter. Consider testing at a level of significance Q.30 (D) be a random sample of size if and only if > < from a discrete distribution with the probability mass function ; = = , = { , where = { . , . } is the unknown parameter. For testing consider a test with the critical region Let and Then (A) (C) MS . . , ={ , { , } { , } + = , = , : = . against : = . , < }. denote the probability of Type I error and power of the test, respectively. is , . , . (B) (D) . . , . , . 10/17 JAM 2018 Mathematical Statistics - MS SECTION - B MULTIPLE SELECT QUESTIONS (MSQ) Q. 31 Q. 40 carry two marks each. Q.31 } Let { be a sequence of real numbers such that = , = + . Then which of the following statement(s) is (are) true? } (A) { (B) { } is bounded below (D) { } is a convergent sequence } (C) { Q.32 is an increasing sequence Let (A) is bounded above be a convergent series of positive real numbers. Then which of the following statement(s) is (are) true? (B) (D) (C) Q.33 Let { } is always convergent is always convergent is always convergent is always convergent / be a sequence of real numbers such that + = Then which of the following statement(s) is (are) true? (A) { (B) { (C) { } } } (D) lim MS = + and, for . , is a monotone sequence is a bounded sequence does not have finite limit, as = 11/17 JAM 2018 Q.34 Mathematical Statistics - MS Let : be defined by = { , + sin , , = . Then which of the following statement(s) is (are) true? (A) attains its minimum at 0 (B) is monotone (C) is differentiable at 0 > (D) Q.35 Let + , for all > be a probability function that assigns the same weight to each of the points of the sample space = { , , , }. Consider the events following statement(s) is (are) true? (A) and are independent (B) and are independent (C) and are independent (D) , Q.36 and , Let = { , } and = { , }. Then which of the are independent , , function = { , }, , , be a random sample from a distribution with the probability density ; = { , , , otherwise, where is the unknown parameter. Then which of the following statement(s) is (are) true? % confidence interval of has to be of finite length (A) A (B) min{ (A) (C) MS } + ln . , min{ , , , } is a % confidence interval of % confidence interval of can be of length (D) A Let , , % confidence interval of can be of length (C) A Q.37 , , , , = max{ = be a random sample from , , , , , where > is the unknown parameter. Let }. Then which of the following is (are) consistent estimator(s) of ? (B) (D) + + 12/17 JAM 2018 Q.38 Mathematical Statistics - MS Let , , , be a random sample from a distribution with the probability density function ; = { , , , otherwise, where is the unknown parameter. Then which of the following statement(s) is (are) true? (A) The maximum likelihood estimator of is (B) = , for all i { , , , } (C) The maximum likelihood estimator of is min{ , , , (D) The maximum likelihood estimator of does not exist Q.39 Let , , , where > be a random sample from a distribution with the probability density function ; = { , is the unknown parameter. If (are) true? (A) (B) (C) (D) Q.40 Let Let (B) (C) (D) MS = = , > , otherwise, , then which of the following statement(s) is is a complete sufficient statistic for is the uniformly minimum variance unbiased estimator of + is the uniformly minimum variance unbiased estimator of is the uniformly minimum variance unbiased estimator of , , , = max{ be a random sample from , , , statement(s) is (are) true? (A) } } and = min{ , , + , , , where is the unknown parameter. }. Then which of the following is a consistent estimator of is a consistent estimator of + is a consistent estimator of is a consistent estimator of 13/17 JAM 2018 Mathematical Statistics - MS SECTION C NUMERICAL ANSWER TYPE (NAT) Q. 41 Q. 50 carry one mark each. Q.41 } Let { Then Q.42 be a sequence of real numbers such that + + + + ! = converges to ____________ : , , Then the area of equals ____________ Q.43 Let = { , Q.44 Let :| | + | | . Then the value of equals ____________ A fair die is rolled three times independently. Given that probability that Let and }. }. Then the area of equals ____________ = Q.46 , Let ={ , Q.45 . appeared at least once, the conditional appeared exactly twice equals ____________ be two positive integer valued random variables with the joint probability mass function where MS = , = , and = = ={ , , , , , otherwise, . Then equals ____________ 14/17 JAM 2018 Q.47 Mathematical Statistics - MS Let , and | Then Q.48 be three events such that | = . , equals ____________ = Let , and be three events such that = , = , , ; | . and ( ) = , and = | . Then the probability that none of the events , , occur equals ____________ Q.49 Let , , , Then = Q.50 Let = be a random sample from the distribution with the probability density function = | | | | + converges in probability to ____________ . , = . = . and = , . . be the observed values of a random sample of size three from a distribution with the probability density function ; = { , / , > , otherwise, where = { , , } is the unknown parameter. Then the maximum likelihood estimate of equals ____________ MS 15/17 JAM 2018 Mathematical Statistics - MS Q. 51 Q. 60 carry two marks each. Q.51 equals ___________ Let = = = [ , where ], = [ lim ( + equals ____________ Let si ] and ) = and lim = = [ . Define ]. = . Then where sin be a random variable with the probability density function = { MS The value of Q.55 = [ Let : be a differentiable function with lim equals ____________ Q.54 ], Then the rank of equals ____________ Q.53 is continuous on with ( ( + ) ( )) . = Then lim Q.52 Let : be a differentiable function such that is a positive integer. Then < , < , , < < , < , otherwise, equals ____________ 16/17 JAM 2018 Q.56 Mathematical Statistics - MS Let and be two discrete random variables with the joint moment generating function + Then Q.57 Let , , Let Then Q.59 Let and + Then Q.58 > , , Then Q.60 + ) ( equals ____________ + ) , , . be i.i.d. discrete random variables with the probability mass function + + = = = { , , = , , , equals ____________ otherwise. be a random variable with the probability mass function = = { max{ , } equals ____________ be a sample observation from parameter. For testing let =( and + , : = , , = , , , otherwise. , distribution, where = { , } is the unknown against : = , be the size and power, respectively, of the test that rejects equals ____________ if and only if . . A fair die is rolled four times independently. For = , , , , define Then max{ , , , }= = { , if appears in the throw, , otherwise. equals ____________ END OF THE QUESTION PAPER MS 17/17 Paper Code : MS Q No Question Type (QT) Section Key/Range 1 MCQ A A 2 MCQ A A 3 MCQ A D 4 MCQ A B 5 MCQ A B 6 MCQ A C 7 MCQ A C 8 MCQ A B 9 MCQ A C 10 MCQ A D 11 MCQ A B 12 MCQ A D 13 MCQ A C 14 MCQ A C 15 MCQ A A 16 MCQ A D 17 MCQ A C 18 MCQ A B 19 MCQ A A 20 MCQ A A 21 MCQ A D 22 MCQ A B 23 MCQ A C Paper Code : MS Q No Question Type (QT) Section Key/Range 24 MCQ A D 25 MCQ A A 26 MCQ A D 27 MCQ A C 28 MCQ A B 29 MCQ A A 30 MCQ A D 31 MSQ B A;B;C;D 32 MSQ B A;C 33 MSQ B A;C 34 MSQ B A;C 35 MSQ B A;C 36 MSQ B B;C;D 37 MSQ B B;C;D 38 MSQ B A 39 MSQ B A;C 40 MSQ B B;C;D 41 NAT C 5.40 to 5.50 42 NAT C 7.80 to 7.90 43 NAT C 1.90 to 2.10 44 NAT C 0.35 to 0.40 45 NAT C 0.16 to 0.17 46 NAT C 2.50 to 3.50 Paper Code : MS Q No Question Type (QT) Section Key/Range 47 NAT C 0.25 to 0.35 48 NAT C 0.30 to 0.40 49 NAT C 4.75 to 5.25 50 NAT C 1.90 to 2.10 51 NAT C 125 to 127 52 NAT C 3.0 to 3.0 53 NAT C 7.25 to 7.50 54 NAT C 1.70 to 1.80 55 NAT C 0.85 to 0.90 56 NAT C 0.97 to 0.99 57 NAT C 0.01 to 0.03 58 NAT C 6.25 to 6.75 59 NAT C 1.10 to 1.20 60 NAT C 0.50 to 0.53

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