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IIT JAM 2020 : Mathematical Statistics

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JAM 2020 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 2020 MATHEMATICAL STATISTICS - MS Special Instructions / Useful Data The set of all real numbers , : , , , 1,2, , , 2, 3, First order derivative of the differentiable function Inverse of the non-singular matrix det( ) trace Determinant of the square matrix Trace of the square matrix Identity matrix of order 2, 3, , Probability of event Complement of an event Expectation of the random variable Var Variance of the random variable i.i.d. Bin , Independently and identically distributed Binomial distribution with parameters 0, 1 0 Poisson( ) Poisson distribution with mean , , 1, 2, and and , , Continuous uniform distribution on the interval , Exponential distribution with probability density function Exp , 0, , Normal distribution with mean and variance Central Chi-squared distribution with Central -distribution with For , Central distribution with ! For 1 3 2 1, ! ! ! , MS 0.6915, 1 1 2 1, 2, , where , degrees of freedom; and , 1, 2, and 0! 0,1, , and 0.5 and 0 and 1, 2, degrees of freedom, 0, 1 and positive integers , 0 degrees of freedom, 0, 1 and positive integer , , , , 0 , otherwise , , 1, 2, , , 1 1, 2, , 0.8413, 1.5 0.9332, 2 0.9772 2/15 JAM 2020 MATHEMATICAL STATISTICS - MS SECTION A MULTIPLE CHOICE QUESTIONS (MCQ) Q. 1 Q.10 carry one mark each. Q.1 Q.2 is a sequence of real numbers such that lim If 0.001, then (A) is a bounded sequence (B) is an unbounded sequence (C) is a convergent sequence (D) is a monotonically decreasing sequence For real constants and , let sin 2 , 0 , If is a differentiable function then the value of (A) 0 Q.3 (B) 1 is (C) 2 (D) 3 2 The area of the region bounded by the curves is (A) Q.4 0 (B) (C) and 2 , , (D) Consider the following system of linear equations 2 5 5 0 1 1 Which one of the following statements is TRUE? 1, (A) The system has unique solution for 1, (B) The system has unique solution for 1, (C) The system has no solution for 1 1 0 0, (D) The system has infinitely many solutions for Q.5 MS Let and TRUE? 0 be two events. Then which one of the following statements is NOT always (A) max 1 (C) min ,0 ,1 (B) max , (D) min , 3/15 JAM 2020 Q.6 MATHEMATICAL STATISTICS - MS be a random variable having Poisson 2 distribution. Then Let (A) 1 Q.7 1 (C) (B) equals (D) The mean and the standard deviation of weights of ponies in a large animal shelter are 20 kg and 3 kg, respectively. A pony is selected at random from the shelter. Using Chebyshev s inequality, the value of the lower bound of the probability that the weight of the selected pony is between 14 kg and 26 kg is Q.8 (C) 0 (B) (A) , Let , , then Var 1 9 and , equals (B) (A) Q.9 1, 2 distribution. If be a random sample from 1 10 (D) 1 (C) (D) 1 and Var be a sequence of i.i.d. random variables such that Let 1, 2, . Then the approximate distribution of 0, 1 (A) , Q.10 Let , , (B) 0, 2 (C) be i.i.d. random variables having 1, , for large , is 0, 0.5 , (D) 0, 0.25 distribution, where and 0. Define 1 2 Then MS , as an estimator of , is (A) biased and consistent (B) unbiased and consistent (C) biased and inconsistent (D) unbiased and inconsistent 4/15 JAM 2020 MATHEMATICAL STATISTICS - MS Q. 11 Q. 30 carry two marks each. Q.11 Let be , (A) a sequence of real numbers 2. Assuming that lim such exists, the value of lim (C) 5 (B) 1, that 7 and is (D) Q.12 Which one of the following series is convergent? 5 4 A B sin C Q.13 Let 1 1 D / and 1 1 cos 1 be two real numbers. If lim then tan 2 2 sin 1 cos 2 equals (B) 1 (A) Q.14 Let : (C) (D) be defined by 2 , Let 1 0, 0 and 3 , , 0, 0 0, , 0, 0 0, 0 denote the first order partial derivatives of , with respect to and , respectively, at the point 0, 0 . Then which one of the following statements is TRUE? (A) is continuous at 0, 0 but (B) is differentiable at 0, 0 (C) is not differentiable at 0, 0 (D) MS 0, 0 and is not continuous at 0, 0 but 0, 0 do not exist 0, 0 and 0, 0 exist 5/15 JAM 2020 MATHEMATICAL STATISTICS - MS Q.15 If the volume of the region bounded by the paraboloid and the plane 2 is given by then (A) 2 and , 0, 2 (B) 1 and (C) 2 and , 0, 2 (D) 1 and , , 0, 1 0, 1 Q.16 The value of the integral is (B) (A) Q.17 Let : 1, 0, 1 (C) be a linear transformation. If 2, 4, 0, 0 and (A) 1, 1, 1, 0 Q.18 Let be an is always (A) singular 0, 1, 1 (B) 0, 1, 1, 1 (D) 1, 1, 0 0, 0, 2, 0 , then (C) 2, 2, 1, 0 2, 0, 0, 0 , 1, 1, 1 equals (D) 0, 0, 0, 0 non-zero skew symmetric matrix. Then the matrix (B) symmetric (C) orthogonal (D) idempotent Q.19 A packet contains 10 distinguishable firecrackers out of which 4 are defective. If three firecrackers are drawn at random (without replacement) from the packet, then the probability that all three firecrackers are defective equals (A) MS (B) (C) (D) 6/15 JAM 2020 MATHEMATICAL STATISTICS - MS Q.20 Let , , , be i.i.d. random variables having a continuous distribution. Then max , equals (B) (A) (C) (D) Q.21 Consider the simple linear regression model , are i.i.d. random variables with mean 0 and variance where 600, , with 500 and 0, . Suppose that 20, 100, 50, 400. Then the least square estimates of are, respectively, (A) 5 and Q.22 Let , , , we have a data set and 1, 2, , , 5 and (B) and (C) 5 and 0, 1 random variables. If be i.i.d. (B) 1 (A) (D) , then (C) 5 and 4 equals (D) Q.23 Consider a sequence of independent Bernoulli trials with probability of success in each trial being . Let denote the number of trials required to get the second success. Then 5 equals (A) (B) , Q.24 Let the joint probability density function of , (A) (B) , , , , MS , 0 otherwise (C) be a random sample from . Then (A) 2 0, be equals Then Q.25 Let (D) (C) (B) (D) 0, 1 distribution and let equals (C) (D) 7/15 JAM 2020 MATHEMATICAL STATISTICS - MS , Q.26 Let , , , be a random sample from , random sample from distribution and distribution, where , , , , 1, 2 and be a 0. Suppose that the two random samples are independent. Define 1 and Then which one of the following statements is TRUE for all positive integers (A) (C) , Q.27 Let (B) , , . If X (D) , , , , , 0.5 , be a random sample from min and ? and X max , 0.5 , , distribution, where , then which one of the following estimators is NOT a maximum likelihood estimator of ? (A) 3 (C) , Q.28 Let , , 1 (B) 3 1 (D) 3 2 be a random sample from Exp , then a 95% confidence interval for A 0, , . B C 0, , . D , Q.29 Let 2 , , be a random sample from random sample from 0, . If distribution, where is , . , , . , 2 1, 2 distribution and let , , , be a 0, 1 distribution. Suppose that the two random samples are independent. Define 1, 0, If lim (A) MS if 1 , otherwise 1, for all (B) 0, then (C) log 1,2, equals (D) log 2 8/15 JAM 2020 MATHEMATICAL STATISTICS - MS Q.30 Let , , , be a random sample from a distribution with probability density function 1 , 0 1 , 0, otherwise To : test 1 size 0 : against 1 would reject 1, the 0 uniformly most powerful test of if (A) log 1 , (B) log 1 , (C) log 1 , (D) log 1 , SECTION - B MULTIPLE SELECT QUESTIONS (MSQ) Q. 31 Q. 40 carry two marks each. Q.31 Let the sequence sin be given by , 1, 2, . Then which of the following statements is/are TRUE? has a subsequence that converges to (A) The sequence (B) lim sup (C) lim inf 1 (D) The sequence Q.32 Let : 1 has a subsequence that converges to be defined by , 3 2 9 , where , . Which of the following is/are saddle point(s) of ? (A) 0, 1 (B) 0, 3 Q.33 The arc length of the parabola (C) MS 1 / (D) 3, 2 2 intercepted between the points of intersection of 2 and the straight line the parabola (A) (C) 3, 2 2 equals (B) (D) 1 4 / 9/15 JAM 2020 MATHEMATICAL STATISTICS - MS Q.34 For real constants and , let 1 1 2 2 be an orthogonal matrix. Then which of the following statements is/are always TRUE? 0 (A) 1 (B) (C) (D) Q.35 Consider a sequence of independent Bernoulli trials with probability of success in each trial being . Then which of the following statements is/are TRUE? (A) Expected number of trials required to get the first success is 5 (B) Expected number of successes in first 50 trials is 10 (C) Expected number of failures preceding the first success is 4 (D) Expected number of trials required to get the second success is 10 , Q.36 Let have the joint probability mass function 1 1 6 0, , 16 5 6 1 2 , 0, 1, , otherwise 1; 0, 1, ,16 Then which of the following statements is/are TRUE? (B) Var (A) Q.37 Let , , be i.i.d. (D) Var (C) 3 0, 1 random variables. Then which of the following statements is/are TRUE? (A) (C) MS (B) , | (D) | 10/15 JAM 2020 MATHEMATICAL STATISTICS - MS Q.38 Let be a sequence of i.i.d. random variables such that 1 0 1 1 4 Define 1 and 1 1 , 1, 2, Then which of the following statements is/are TRUE? (A) lim 1 (C) lim Q.39 Let , , , 1 (B) lim 2 be i.i.d. Poisson 1 0 (D) lim 1 0. Define random variables, where 1 and 1 Then which of the following statements is/are TRUE? (A) Var Var (B) Var Var (C) Var attains the Cramer-Rao lower bound (D) Q.40 Consider the following two probability density functions (pdfs) 2 , if 0 1 and 0, otherwise Let : be a random variable having pdf , 0, 1 against 1, if 0 1 0, otherwise , . Consider testing , 0, 1 at : 0.05 level of significance. For which of the following observed values of random observation most powerful test would reject (A) 0.19 MS (B) 0.22 , the ? (C) 0.25 (D) 0.28 11/15 JAM 2020 MATHEMATICAL STATISTICS - MS SECTION C NUMERICAL ANSWER TYPE (NAT) Q. 41 Q. 50 carry one mark each. Q.41 lim 1 Q.42 1 equals _________ , The maximum value of the function , is __________ Q.43 The value of the integral equals __________ (round off to two decimal places) Q.44 The rank of the matrix 1 1 2 2 is __________ Q.45 Let , Then the conditional expectation places) that 1 1 3 2 6 4 8 5 have the joint probability density function , Q.46 Let 1 2 5 6 3 , 0 2 4 0, otherwise | 1 equals __________ (round off to two decimal be a random variable having the Poisson 4 distribution and let be an event such | 1 2 , 0, 1, 2, . Then equals __________ (round off to two decimal places) MS 12/15 JAM 2020 MATHEMATICAL STATISTICS - MS , Q.47 Let and be independent random variables such that 55,15 and 60, 14 . Then , 47, 10 , equals __________ (round off 2 to two decimal places) Q.48 Let , and 3.69 . If 0.05, then the value of such that 0.95 equals __________ (round off to two decimal places) Q.49 Let the sample mean based on a random sample from Exp distribution be 3.7. Then the equals __________ (round off to two decimal maximum likelihood estimate of 1 places) Q.50 Let be a single observation drawn from : 1 against : 0, distribution, where 1, 2 . To test 2 consider the test procedure that rejects 0.75. If the probabilities of Type-I and Type-II errors are if and only if and , respectively, then equals __________ (round off to two decimal places) Q. 51 Q. 60 carry two marks each. Q.51 Let | Q.52 Let : 1, 3 | , 1, 3 , 1 1 and 3 ! . Then is differentiable on 7. Then be the real number such that the coefficient of is MS be a continuous function such that 1, 3 , 1 equals __________ in Maclaurin s series of equals __________ 13/15 JAM 2020 MATHEMATICAL STATISTICS - MS Q.53 Consider the matrix Let 1 0 0 0 0 3 2 1 4 be a non-singular matrix such that equals __________ matrix Q.54 Let is a diagonal matrix. Then the trace of the be a 3 3 matrix having characteristic roots 3 and 3 0 and , 2 . If det and 1. Define trace , then equals __________ (round off to two decimal places) Q.55 Let and be independent random variables with respective moment generating functions 8 81 3 64 , 1 equals __________ (round off to two decimal places) Then Q.56 Let 1 and be a random variable having 0,10 distribution and denotes the greatest integer less than or equal to . Then , where 0.25 equals __________ Q.57 A computer lab has two printers handling certain types of printing jobs. Printer-I and Printer-II handle 40% and 60% of the jobs, respectively. For a typical printing job, printing time (in minutes) of Printer-I follows follows 10, 4 distribution and that of Printer-II 1, 21 distribution. If a randomly selected printing job is found to have been completed in less than 10 minutes, then the conditional probability that it was handled by the Printer-II equals __________ (round off to two decimal places) MS 14/15 JAM 2020 MATHEMATICAL STATISTICS - MS Q.58 Let 1, 0, 6 from Bin 1, 0, 1, 0, distribution, where 1 unbiased estimate of Q.59 Let 1, 1 be the data on a random sample of size 0, 1 . Then the uniformly minimum variance equals __________ 4 be the data on a random sample of size 2 from a Poisson distribution, where 0, . Let estimate of ! be the uniformly minimum variance unbiased based on the given data. Then equals __________ , 1 distribution, where . Consider testing (round off to two decimal places) Q.60 Let : be a random variable having 0 against : likelihood ratio test at 0 at 0.617 level of significance. The power of the 1 equals __________ (round off to two decimal places) END OF THE QUESTION PAPER MS 15/15

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