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GATE 2019 : Statistics (with Final Answer Keys)

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GATE 2019 General Aptitude (GA) Set-8 Q. 1 Q. 5 carry one mark each. Q.1 The fishermen, _______ the flood victims owed their lives, were rewarded by the government. (A) whom Q.2 (B) to which (C) to whom (D) that Some students were not involved in the strike. If the above statement is true, which of the following conclusions is/are logically necessary? 1. 2. 3. 4. Some who were involved in the strike were students. No student was involved in the strike. At least one student was involved in the strike. Some who were not involved in the strike were students. (A) 1 and 2 Q.3 (C) 4 (D) 2 and 3 The radius as well as the height of a circular cone increases by 10%. The percentage increase in its volume is ______. (A) 17.1 Q.4 (B) 3 (B) 21.0 (C) 33.1 (D) 72.8 Five numbers 10, 7, 5, 4 and 2 are to be arranged in a sequence from left to right following the directions given below: 1. No two odd or even numbers are next to each other. 2. The second number from the left is exactly half of the left-most number. 3. The middle number is exactly twice the right-most number. Which is the second number from the right? (A) 2 Q.5 (C) 7 (D) 10 Until Iran came along, India had never been ________________ in kabaddi. (A) defeated GA (B) 4 (B) defeating (C) defeat (D) defeatist 1/3 GATE 2019 General Aptitude (GA) Set-8 Q. 6 Q. 10 carry two marks each. Q.6 Since the last one year, after a 125 basis point reduction in repo rate by the Reserve Bank of India, banking institutions have been making a demand to reduce interest rates on small saving schemes. Finally, the government announced yesterday a reduction in interest rates on small saving schemes to bring them on par with fixed deposit interest rates. Which one of the following statements can be inferred from the given passage? (A) Whenever the Reserve Bank of India reduces the repo rate, the interest rates on small saving schemes are also reduced (B) Interest rates on small saving schemes are always maintained on par with fixed deposit interest rates (C) The government sometimes takes into consideration the demands of banking institutions before reducing the interest rates on small saving schemes (D) A reduction in interest rates on small saving schemes follow only after a reduction in repo rate by the Reserve Bank of India Q.7 In a country of 1400 million population, 70% own mobile phones. Among the mobile phone owners, only 294 million access the Internet. Among these Internet users, only half buy goods from e-commerce portals. What is the percentage of these buyers in the country? (A) 10.50 Q.8 (B) 14.70 (C) 15.00 (D) 50.00 The nomenclature of Hindustani music has changed over the centuries. Since the medieval period dhrupad styles were identified as baanis. Terms like gayaki and baaj were used to refer to vocal and instrumental styles, respectively. With the institutionalization of music education the term gharana became acceptable. Gharana originally referred to hereditary musicians from a particular lineage, including disciples and grand disciples. Which one of the following pairings is NOT correct? (A) dhrupad, baani (B) gayaki, vocal (C) baaj, institution (D) gharana, lineage Q.9 Two trains started at 7AM from the same point. The first train travelled north at a speed of 80km/h and the second train travelled south at a speed of 100 km/h. The time at which they were 540 km apart is ______ AM. (A) 9 GA (B) 10 (C) 11 (D) 11.30 2/3 GATE 2019 General Aptitude (GA) Set-8 Q.10 I read somewhere that in ancient times the prestige of a kingdom depended upon the number of taxes that it was able to levy on its people. It was very much like the prestige of a head-hunter in his own community. Based on the paragraph above, the prestige of a head-hunter depended upon ___________ (A) the prestige of the kingdom (B) the prestige of the heads (C) the number of taxes he could levy (D) the number of heads he could gather END OF THE QUESTION PAPER GA 3/3 GATE 2019 STATISTICS Q. 1 Q. 25 carry one mark each. Q.1 lim =1 2 + 2 is equal to (A) 3 Q.2 5 6 (C) 3 4 (D) 4 Let = ( + )( + ) be a vector field on 3 . The line integral , where is the triangle with vertices (0,0,0), (5,0,0) and (5,5,0) traversed in that order is (A) -25 Q.3 (B) (B) 25 (C) 50 (D) 5 Let {1,2,3,4} represent the outcomes of a random experiment, and ({1}) = ({2}) = ({3}) = ({4}) = 1/4. Suppose that 1 = {1,2}, 2 = {2,3}, 3 = {3,4}, and 4 = {1,2,3}. Then which of the following statements is true? (A) 1 and 2 are not independent. (B) 3 and 4 are independent. (C) 1 and 4 are not independent. (D) 2 and 4 are independent. Q.4 A fair die is rolled two times independently. Given that the outcome on the first roll is 1, the expected value of the sum of the two outcomes is (A) 4 Q.5 (B) 28 (C) 27 (D) 26 (B) 2 or 3 (C) 4 (D) 5 Let 1 , , be a random sample from uniform distribution defined over (0, ), where > 0 and 2. Let (1) = min{ 1 , , } and ( ) = max{ 1 , , }. Then the covariance between ( ) and (1) / ( ) is (A) 0 ST (D) 5.5 Let be a 6 6 complex matrix with 3 and 4 = . Then the number of Jordan blocks of is (A) 1 or 6 Q.7 (C) 3 The dimension of the vector space of 7 7 real symmetric matrices with trace zero and the sum of the off-diagonal elements zero is (A) 47 Q.6 (B) 4.5 (B) ( + 1) (C) (D) 2 ( + 1) 1/10 GATE 2019 Q.8 STATISTICS Let 1 , , be a random sample drawn from a population with probability density function ( ; ) = 1 , 0 1, > 0. Then the maximum likelihood estimator of is (A) =1 log Q.9 (B) =1 log (C) ( =1 )1/ (D) =1 Let = 0 + 1 1 + 2 2 + , for = 1, ,10, where 1 s and 2 s are fixed covariates, and s are uncorrelated random variables with mean 0 and unknown variance 2 . Here 0 , 1 and 2 are unknown parameters. Further, define = 0 + 1 1 + 2 2 , where ( 0 , 1 , 2 ) is the unbiased least squares estimator of ( 0 , 1 , 2 ). Then an unbiased estimator of 2 is (A) (C) 2 10 =1( ) (B) 10 2 10 =1( ) (D) 8 2 10 =1( ) 7 2 10 =1( ) 9 Q.10 For = 1,2,3, let = + + , where s are fixed covariates, and s are independent and identically distributed standard normal random variables. Here, and are unknown parameters. Given the following observations, 0.5 1 2.5 1 0.5 -2 the best linear unbiased estimate of + is equal to (A) 1.5 (B) 1 (C) 1.8 (D) 2.1 Q.11 Consider a discrete time Markov chain on the state space {1,2,3} with one-step transition 1 2 3 1 0.7 0.3 0 probability matrix . Which of the following statements is true? 2 [ 0 0.6 0.4] 3 0 0 1 (A) States 1, 3 are recurrent and state 2 is transient. (B) State 3 is recurrent and states 1, 2 are transient. (C) States 1, 2, 3 are recurrent. (D) States 1, 2 are recurrent and state 3 is transient. Q.12 1 The minimal polynomial of the matrix [0 0 0 1 2 0 0 2 1 1 0 0 0] is 0 2 (A) ( 1)( 2) (B) ( 1)2 ( 2) (C) ( 1)( 2)2 (D) ( 1)2 ( 2)2 ST 2/10 GATE 2019 STATISTICS Q.13 Let ( 1 , 2 , 3 ) be a trivariate normal random vector with mean vector ( 3, 1,4) and 4 0 0 variance-covariance matrix [0 3 3]. Which of the following statements are true? 0 3 4 (i) 2 and 3 are independent. (ii) 1 + 3 and 2 are independent. (iii) ( 2 , 3 ) and 1 are independent. 1 (iv) ( 2 + 3 ) and 1 are independent. 2 (A) (i) and (iii) (B) (ii) and (iii) (C) (i) and (iv) (D) (iii) and (iv) Q.14 A 23 factorial experiment with factors , and is arranged in two blocks of four plots each as follows: (Below (1) denotes the treatment in which , and are at the lower level, ac denotes the treatment in which and are at the higher level and is at the lower level and so on.) Block 1 Block 2 (1) a ab b ac c bc abc The treatment contrast that is confounded with the blocks is (A) (B) (C) (D) Q.15 Consider a fixed effects two-way analysis of variance model = + + + + , where = 1, , ; = 1, , ; = 1, , , and s are independent and identically distributed normal random variables with zero mean and constant variance. Then the degrees of freedom available to estimate the error variance is zero when (A) = 1 (B) = 1 (C) = 1 (D) None of the above. Q.16 For = 1,2, ,10, let the probability density function of the random variable be ( ) = { , 0, >0 . Then ( 10 =1 ) is equal to Q.17 The probability density function of the random vector ( , ) is given by , , ( , ) = { 0, 0< < <1 . Then the value of is equal to .. ST 3/10 GATE 2019 STATISTICS Q.18 Let { } 1 be a sequence of independent and identically distributed normal random 1 variables with mean 4 and variance 1. Then lim ( =1 > 4.0006) is equal to Q.19 Let ( 1 , 2 ) be a random vector following bivariate normal distribution with mean vector (0, 0), Variance( 1 ) = Variance( 2 ) = 1 and correlation coefficient , where | | < 1. Then ( 1 + 2 > 0) is equal to Q.20 Let 1 , , be a random sample from normal distribution with mean and variance 1. Let be the cumulative distribution function of the standard normal distribution. Given (1.96) = 0.975, the minimum sample size required such that the length of the 95% confidence interval for does NOT exceed 2 is Q.21 Let be a random variable with probability density function ( ; ) = , where 0 and > 0. To test 0 : = 1 against 1 : > 1, the following test is used: Reject 0 if and only if > log 20. Then the size of the test is Q.22 Let { } 0 be a discrete time Markov chain on the state space {1,2,3} with one-step transition probability matrix 1 2 3 1 0.4 0.3 0.3 2 [0.5 0.2 0.3 ] 3 0.2 0.4 0.4 and initial distribution ( 0 = 1) = 0.5, ( 0 = 2) = 0.2, ( 0 = 3) = 0.3 . Then ( 1 = 2, 2 = 3, 3 = 1) (rounded off to three decimal places) is equal to Q.23 Let be a continuous and positive real valued function on [0, 1]. Then 0 (sin ) cos is equal to Q.24 A random sample of size 100 is classified into 10 class intervals covering all the data points. To test whether the data comes from a normal population with unknown mean and unknown variance, the chi-squared goodness of fit test is used. The degrees of freedom of the test statistic is equal to ST 4/10 GATE 2019 STATISTICS Q.25 For = 1,2,3,4, let = + + where s are fixed covariates and s are uncorrelated random variables with mean 0 and variance 3. Here, and are unknown parameters. Given the following observations, 2 3 2.5 2 -0.5 -4 1 -1 the variance of the least squares estimator of is equal to ST 5/10 GATE 2019 STATISTICS Q. 26 Q. 55 carry two marks each. Q.26 Let = ( 1) +1 , 0, and = , 0. Then, for | | < 1, the series =0 ! =0 converges to (A) (B) 1+ 1 2 (C) 1 (D) (1 + ) Q.27 Let { } 1 be a sequence of independent and identically distributed Bernoulli random variables with success probability (0,1). Then, as , 1 ( ) =1 converges almost surely to 1 (A) (B) 1 (C) 1 (D) 1 2 Q.28 Let and be two independent random variables with and 2 distributions, respectively, where and are positive integers. Then which of the following statements is true? (A) For < , ( > ) ( > ) for all . (B) For > , ( > ) ( > ) for all . (C) For < , ( > ) = ( > ) for all . (D) None of the above. Q.29 1 The matrix [0 2 0 0 ( , , ) equals (A) (0,0,1) ] is diagonalizable when 1 (B) (1,1,0) (C) ( 2, 2, 2) (D) ( 2, 2, 2) Q.30 Suppose that 1 and 2 are two populations with equal prior probabilities having bivariate normal distributions with mean vectors (2, 3) and (1, 1), respectively. The variancecovariance matrix of both the distributions is the identity matrix. Let 1 = (2.5, 2) and 2 = (2, 1.5) be two new observations. According to Fisher s linear discriminant rule, (A) 1 is assigned to 1 , and 2 is assigned to 2 . (B) 1 is assigned to 2 , and 2 is assigned to 1 . (C) 1 is assigned to 1 , and 2 is assigned to 1 . (D) 1 is assigned to 2 , and 2 is assigned to 2 . Q.31 Let 1 , , be a random sample from a population having probability density function 2 ( ; ) = 2 , 0 < < . Then the method of moments estimator of is (A) ST 3 =1 2 (B) 2 3 =1 2 (C) =1 (D) 3 =1 ( 1) 2 6/10 GATE 2019 STATISTICS Q.32 Let be a normal random variable having mean and variance 1, where 1 10. Then is (A) sufficient but not complete. (B) the maximum likelihood estimator of . (C) the uniformly minimum variance unbiased estimator of . (D) complete and ancillary. Q.33 Let { } 1 be a sequence of independent and identically distributed random variables with mean and variance , where > 0. Then 1 (A) 1+ (B) =1 =1 2 is a consistent estimator of 1 1+ (C) (D) 1+ Q.34 Let 1 , , 10 be a random sample from a population with probability density function ( ; ) = | | , < < , < < . 2 Then the maximum likelihood estimator of (A) does not exist. (B) is not unique. (C) is the sample mean. (D) is the smallest observation. Q.35 Consider the model = + , where s are independent normal random variables with zero mean and known variance 2 > 0, for = 1, , . Then the best linear unbiased estimator of the unknown parameter is (A) 2 =1( / ) 2 =1(1/ ) (B) =1 (C) =1( / ) (D) =1( / ) =1(1/ ) Q.36 Let ( , ) be a bivariate random vector with probability density function ( , ) ( , ) = { , 0, 0 < < , . Then the regression of on is given by (A) + 1 ST (B) 2 (C) 2 (D) + 1 7/10 GATE 2019 STATISTICS Q.37 Consider a discrete time Markov chain on the state space {1, 2} with one-step transition probability matrix 1 2 = 1 0.2 0.8 . [ ] 2 0.3 0.7 Then lim is (A) 3 8 [11 3 11 8] 11 11 (B) [ 1 0 ] 0 1 (C) [ 0 1 ] 1 0 (D) 8 3 [11 8 11 3] 11 11 Q.38 Let ( , ) be a random vector with variance-covariance matrix [4 0]. The two 1 2 0 2 principal components are (A) 1 and 2 (B) 1 and 2 (C) 1 and 2 (D) 1 + 2 and 2 Q.39 Consider the objects {1,2,3,4} with the distance matrix 1 2 3 4 0 1 11 5 1 2 [1 0 2 3] . 3 11 2 0 4 4 5 3 4 0 Applying the single-linkage hierarchical procedure twice, the two clusters that result are (A) {2,3} and {1,4} (C) {1,3,4} and {2} (B) {1,2,3} and {4} (D) {2,3,4} and {1} Q.40 The maximum likelihood estimates of the mean vector and the variance-covariance matrix 4 4 1 of a bivariate normal distribution based on the realization {( ) , ( ) , ( ) } of a random 3 2 4 sample of size 3, are given by 3 (A) ( ) and 3 3 (C) ( ) and 3 2 1 ] 1 2/3 3 3/2 [ ] 3/2 2/3 [ 3 (B) ( ) and 3 3 (D) ( ) and 3 2 1 ] 1 3/2 3 2/3 [ ] 2/3 1 [ Q.41 Consider a fixed effects one-way analysis of variance model = + + , for = 1, , , = 1, , , and s are independent and identically distributed normal random variables with mean zero and variance 2 . Here, and are positive integers. Let = (A) + ST 2 =1 . Then is the least squares estimator for (B) (C) + (D) 8/10 GATE 2019 STATISTICS Q.42 Let be a positive semi-definite matrix with eigenvalues 1 , and with as the maximum diagonal entry. We can find a vector such that = 1, where denotes the transpose, and (A) > 1 (B) < (C) 1 (D) > Q.43 Let be a random variable with uniform distribution on the interval ( 1,1) and = ( + 1)2 . Then the probability density function ( ) of , over the interval (0,4), is (A) 3 16 (B) 4 1 (C) 1 (D) 6 1 Q.44 Let be the solid whose base is the region in the -plane bounded by the curves = 2 and = 8 2 , and whose cross-sections perpendicular to the -axis are squares. Then the volume of (rounded off to two decimal places) is Q.45 Consider the trinomial distribution with the probability mass function 7! ( = , = ) = ! !(7 )! (0.6) (0.2) (0.2)7 , 0, 0, and + 7. Then ( | = 3) is equal to Q.46 Let = + + , where = 1, 2, 3, 4, s are fixed covariates and s are independent and identically distributed standard normal random variables. Here, and are unknown parameters. Let be the cumulative distribution function of the standard normal distribution and (1.96) = 0.975. Given the following observations, 3 1 -2.5 -2 5 3 -5 -2 the length (rounded off to two decimal places) of the shortest 95% confidence interval for based on its least squares estimator is equal to Q.47 Consider a discrete time Markov chain on the state space {1,2,3} with one-step transition 1 2 3 1 0 0.2 0.8 probability matrix . Then the period of the Markov chain is 2 [0.5 0 0.5] 3 0.6 0.4 0 Q.48 Suppose customers arrive at an ATM facility according to a Poisson process with rate 5 customers per hour. The probability (rounded off to two decimal places) that no customer arrives at the ATM facility from 1:00 pm to 1:18 pm is ST 9/10 GATE 2019 STATISTICS Q.49 Let be a random variable with characteristic function ( ) such that (2 ) = 1. Let denote the set of integers. Then ( ) is equal to Q.50 Let 1 be a random sample of size 1 from uniform distribution over ( , 2 ), where > 1. To test 0 : = 2 against 1 : = 3, reject 0 if and only if 1 > 3.5 . Let and be the size and the power, respectively, of this test. Then + (rounded off to two decimal places) is equal to Q.51 Let = 0 + 1 + , = 1, , , where s are fixed covariates, and s are uncorrelated random variables with mean zero and constant variance. Suppose that 0 and 1 are the least squares estimators of the unknown parameters 0 and 1, respectively. If =1 = 0, then the correlation between 0 and 1 is equal to Q.52 Let : be defined by ( ) = (3 2 + 4) cos . Then lim ( )+ ( ) 8 2 0 is equal to Q.53 The maximum value of ( 1)2 + ( 2)2 subject to the constraint 2 + 2 45 is equal to Q.54 Let 1 , , 10 be independent and identically distributed normal random variables with 2 1 mean 0 and variance 2. Then ( 2 + + 2 ) is equal to 1 10 Q.55 Let be the 4 4 identity matrix and = (1, 2, 3, 4) , where denotes the transpose. Then the determinant of + is equal to END OF THE QUESTION PAPER ST 10/10

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