**GATE-2020**

**ST: Statistics**

**GA-General Aptitude**

**Q1 – Q5 carry one mark each.**

1. Rajiv Gandhi Khel Ratna Award was conferred _____Mary Kom, a six-time world champion in boxing, recently in a ceremony _____ the Rashtrapati Bhawan (the President’s official residence) in New Delhi.

(A) with, at

(B) on, in

(C) on, at

(D) to, at

2. Despite a string of a poor performances, the changes of K. L. Rahul’s selection in the team are ______.

(A) slim

(B) bright

(C) obvious

(D) uncertain

3. Select the word that fits the analogy:

Cover : Uncover :: Associate : _______

(A) Unassociate

(B) Inassociate

(C) Missassociate

(D) Dissociate

4. Hig by floods, he kharif (summer sown) crops in various parts of the county have been affected. Officials believe that the loss in production of the kharif crops can be recovered in the output of the rabi (winter sown) crops so that the country can achieve its food-grain production target of 291 million tons in the crop year 2019-20 (July-June). They are hopeful that good rains in July-August will help the soil retain moisture for a longer period, helping winter sown crops such as wheat and pulses during the November-February period.

Which of the following statements can be inferred from the given passage?

(A) Officials declared that the food-grain production target will be met due to good rains.

(B) Officials want the food-grain production target to be met by the November-February period.

(C) Officials feel that the food-grain production target cannot be met due to floods.

(D) Officials hope that the food-grain production target will be met due to a good rabi produce.

5. The difference between the sum of the first 2n natural numbers and the sum of the first n odd natural numbers is ______.

(A) n^{2} – n

(B) n^{2} + n

(C) 2n^{2} – n

(D) 2n^{2} + n

**Q6 – Q10 carry two marks each.**

6. Repo rate is the at which Reserve Bank of India (RBI) lends commercial banks, and reverse repo rate is the rate at which RBI borrows money from commercial banks.

Which of the following statements can be inferred from the above passage?

(A) Decrease in repo rate will increase cost of borrowing and decrease lending by commercial banks.

(B) Increase in repo rate will decrease cost of borrowing and increase lending by commercial banks.

(C) Increase in repo rate will decrease cost of borrowing and decrease lending by commercial banks.

(D) Decrease in repo rate will decrease cost of borrowing and increase lending by commercial banks.

7. P, Q, R, S, T, U, V, and W are seated around a circular table.

(I) S is seated opposite to W.

(II) U is seated at the second place to the right of R.

(III) T is seated at the third place to the left of R.

(IV) V is a neighbor of S.

Which of the following must be true?

(A) P is a neighbor of R.

(B) Q is a neighbor of R.

(C) P is not seated opposite to Q.

(D) R is the left neighbor of S.

8. The distance between Delhi and Agra is 233 km. A car P started travelling from Delhi to Agra and another car Q started from Agra to Delhi along the same road 1 hour after the car P started. The two cars crossed each other 75 minutes after the car Q started. Both cars were travelling at constant speed. The speed of car P was 10 km/hr more than the speed of car Q. How many kilometers the car Q had travelled when the cars crossed each other?

(A) 66.6

(B) 75.2

(C) 88.2

(D) 116.5

9. For a matrix M = [m_{ij}], i. j= 1, 2, 3, 4, the diagonal elements are all zero and m_{ij} = −m_{ij}. The minimum number of elements required to fully specify the matrix is_______.

(A) 0

(B) 6

(C) 12

(D) 16

10. The profit shares of two companies P and Q are shown in the figure. If the two companies have invested a fixed and equal amount every year, then the ratio of the total revenue of company P to the total revenue of company Q, during 2013-2018 is ______.

(A) 15 : 17

(B) 16 : 17

(C) 17 : 15

(D) 17 : 16

**ST: Statistics**

**Q1 – Q25 carry one mark each.**

1. Let M be a 3 × 3 non-zero idempotent matrix and let I_{3} denote the 3 × 3 identity matrix. Then which of the following statements is FLASE?

(A) The eigenvalues of M are 0 and 1

(B) Rank(M) = Trace(M)

(C) I_{3} – M is idempotent

(D) (I_{3} + M)^{−}^{1} = I_{3} – 2M

2. Let ℂ denote the set of all complex numbers. Consider the vector space

over the field of real numbers, where for any complex number z, denotes its complex conjugate. If i = √−1, then a basis of V is

(A) {(1, −1, 1), (i, i, i)}

(B) {(1, −1, 1), (i, −i, i)}

(C) {(1, −i, 1), (i, 1, i)}

(D) {(1, −i, 1), (i, 1, −i)

3. Let S = {(x, y) ∈ ℝ × ℝ: x^{2} – y^{2} = 4} and f : S → ℝ be defined by f(x, y) = 6x + y^{2}, where ℝ denotes the set of all real numbers. Then

(A) f is bounded of S

(B) the maximum value of f on S is 13

(C) the minimum value of f on S is −14

(D) the minimum value of f on S is −13

4. Let f : ℝ × ℝ → ℝ be defined by

where ℝ denotes the set of all real numbers and c ∈ ℝ is fixed constant. Then, which of the following statements is TRUE?

(A) There does NOT exists a value of c for which f is continuous at (0, 0)

(B) f is continuous at (0, 0) if c = 0

(C) f is continuous at (0, 0) if c = 10

(D) f is continuous at (0 0) if c = 16

5. The moment generating function of a random variable X is given by

Then P(X ≤ 2) equals

(A) 1/3

(B) 1/6

(C) 1/2

(D) 5/6

6. Consider the following two-way fixed effects analysis of variance model

Y_{ijk} = μ + α_{i} + β_{j} + ϵ_{ijk}, i = 1, 2; j = 1, 2, 3; k = 1, 2, 3;

where ϵ_{ijk}’s are independently and identically distributed N(0, σ^{2}) random variables, σ ∈ (0, ∞), α_{1} + α_{2} = 0 and β_{1} + β_{2} + β_{3} = 0. Let SSE denote the sum of squares due to error. For any positive integer v and any α ∈ (0, 1), let χ^{2}_{v,α} denote the (1 – α)-th quantile of the central chi-square distribution with v degrees of freedom. Then a 95% confidence interval for σ^{2} is given by

7. Let X_{1}, …, X_{20} be independent and identically distributed random variables with the common probability density function

Then the distribution of the random variableis

(A) central chi-square with 10 degrees of freedom

(B) central chi-square with 20 degrees of freedom

(C) central chi-square with 30 degrees of freedom

(D) central chi-square with 40 degrees of freedom

8. Let X_{1}, …, X_{10} be a random sample from a Weibull distribution with the probability density function

where θ ∈ (0, ∞). For any positive integer v and any α ∈ (0, 1), let χ^{2}_{v,}_{α} denote the (1 – α)-th quantile of the central chi-square distribution with v degrees of freedom. Then, a 90% confidence interval for θ is

9. Let X_{1}, …, X_{n} be a random sample of size n (≥2) from a uniform distribution on the interval [−θ, θ], where θ ∈ (0, ∞). A minimal sufficient for θ is

10. Let X_{1} …, X_{n} be a random sample of size n(≥2) from N(θ, 2θ^{2}) distribution, where θ ∈ (0, ∞). Which of the following statements is TRUE?

(A) is the unique unbiased estimator of θ^{2} that is a function of minimal sufficient statistic

(B) is an unbiased estimator of θ^{2}

(C) There exist infinite number of unbiased estimators of θ^{2} which are functions of minimal sufficient statistic

(D) There does NOT exist any unbiased estimator of θ(θ + 1) that is a function of minimal sufficient statistic

11. Let {N(t), t ≥ 0} be a Poisson process with rate λ = 2. Given that N(3) = 1, the expected arrival time of the first event of the process is

(A) 1

(B) 3/2

(C) 2/3

(D) 3

12. Consider the regression model

Y_{i} = β_{0} + β_{1}x_{i}^{2} + ϵ_{i}, i = 1, 2, …, n(n ≥ 2);

where β_{0} and β_{1} are unknown parameters and ϵ_{i}^{’}s are random errors. Let y_{i} be the observed value of Y_{i}, i = 1, …, n. Using the method of ordinary least squares, the estimate of β_{1} is

13. Let be a random sample of size n (≥2) from distribution, where 1 ≤ p ≤ n – 1 and ∑ is a positive definite matrix. Define

where for any column vector denotes its transpose. Then the distribution of the statistic is

(A) χ^{2}_{p}, the central chi-square distribution with p degrees of freedom

(B) F_{p,n}_{−}_{p}, the central F distribution with p and n – p degrees of freedom

(C) where F_{p,n}_{−}_{p}, is the central F distribution with p and n – p degrees of freedom

(D) where F_{p,n}_{−}_{p}, is the central F distribution with n – p and p degrees of freedom

14. Consider a two-way fixed effects analysis of variance model without interaction effect and one observation per cell. If there are 5 factors and 4 columns, then the degrees of freedom for the error sum of squares is

(A) 20

(B) 19

(C) 12

(D) 11

15. Let X_{1}, …, X_{n} be a random sample of size n (≥ 2) from an exponential distribution with the probability density function

where θ ∈ {1, 2}. Consider the problem of testing H_{0 : }θ = 1 against H_{1} : θ = 2, based on X_{1}, …, X_{n}. Which of the following statements is TRUE?

(A) Likelihood ratio test at level α (0 < α < 1) leads to the same critical region as the corresponding most powerful test at the same level.

(B) Critical region of level α (0 < α < 1) likelihood ratio test is is the α-th quantile of the central chi-square distribution with 2n degrees of freedom

(C) Likelihood ratio test for testing H_{0} against H_{1} does not exist

(D) At any fixed level α (0 < α < 1), the power of the likelihood ratio test is lower than that of the most powerful test

16. Te characteristic function of a random variable X is given by

Then P(|X| ≤ 3/2) = _____ (correct up to two decimal places).

17. Let the random follow distribution, where

Then P(X_{1} + X_{2} + X_{3} + X_{4} > 0) = _______ (correct up to one decimal place).

18. Let {X_{n}}_{n}_{≥}_{0} be a homogeneous Markov chain with state space {0, 1} and one-step transition probability matrix If P(X_{0} = 0) = 1/3, then 27 × E(X_{2}) = ______ (correct up to two decimal places).

19. Let E, F and G be mutually independent events with P(E) = 1/2, P(F) = 1/3 and P(G) = 1/4. Let p be the probability that at least two of the events among E, F and G occur. Then 12 × p = _______ (correct up to one decimal place).

20. Let the joint probability mass function of (X, Y, Z) be

where k = 10 – x – y – z; x, y, z = 0, 1, … , 10; x + y + z ≤ 10. Then the variance of the random Y + Z equals ______ (correct up to one decimal place).

21. The total number of standard 4 × 4 Latin squares is _______

22. Let be a 4 × 1 random vector with and variance-covariance matrix

Let be the 4 × 1 random vector of principal components derived from ∑. The proportion of total variation explained by the first two principal components equals ______ (correct up to two decimal places).

23. Let X_{1}, …, X_{n} be a random sample of size n (≥ 2) from an exponential distribution with the probability density function

where θ ∈ (0, ∞). If X_{(1)} = min{X_{1}, …, X_{n}} then the conditional expectation

24. Let Y_{i} = α + βx_{i} + ϵ_{i}, i = 1, 2, …, 7, where x_{i}’s are fixed covariates and ϵ_{i}’s are independent and identically distributed random variables with mean zero and finite variance. Suppose that are the least squares estimators of α and β, respectively. Given the following data:

where y_{i} is the observed value of Y_{i}, i = 1, …, 7. Then the correlation coefficient between equals_____

25. Let {0, 1, 2, 3} be an observed sample of size from N(θ, 5) distribution, where θ ∈ [2, ∞). Then the maximum likelihood estimate of θ based on the observed sample is ________.

**Q26 – Q55 carry two marks each.**

26. Let f : ℝ × ℝ → ℝ be defined by

f(x, y) = x^{4} – 2x^{3}y + 16y + 17,

where ℝ denotes the set of all real numbers. Then

(A) f has a local minimum at (2, 4/3)

(B) f has a local maximum at (2, 4/3)

(C) f has a saddle point at (2, 4/3)

(D) f is bounded

27. Consider the linear transformation T : ℂ^{3} = ℂ × ℂ × ℂ. Which of the following statements is TRUE?

(A) There exists a non-zero vector X such that T(X) = −X

(B) There exist a non-zero vector Y and a real number λ ≠ 1 such that T(Y) = λY

(C) T is diagonalizable

(D) T^{2} = I_{3}, where I_{3} is the 3 × 3 identity matrix

28. For real numbers a, b and c, let

Then, which of the following statements is TRUE?

(A) Rank (M) = 3 for every, a, b, c ∈ ℝ

(B) If a + c = 0 then M is diagonalizable for every b ∈ ℝ

(C) M has a pair of orthogonal eigenvectors for every a, b, c ∈ ℝ

(D) If b= 0 and a + c = 1 then M is NOT idempotent

29. Let M be a 4 × 4 matrix with (x – 1)^{2} (x – 3)^{2} as its minimal polynomial. Then, which of the following statements is FALSE?

(A) The eigenvalues of M are 1 and 3

(B) The algebraic multiplicity of the eigenvalue 1 is 3

(C) M is NOT diagonalizable

(D) Trace(M) = 8

30. Let f : ℝ × ℝ → ℝ be defined by

where ℝ denotes the set of all real numbers. Then which of the following statements is TRUE?

(A) f is differentiable at (1, 2)

(B) f is continuous at (1, 2) but NOT differentiable at (1, 2)

(C) The partial derivative of f, with respect to x, at (1, 2) does NOT exist

(D) The directional derivative of f at (1, 2) along equals 1

31. Which of the following functions is uniformly continuous on the specified domain?

32. Let the random vector have the joint probability density function

Which of the following statements is TRUE?

(A) X_{1}, X_{2} and X_{3} are mutually independent

(B) X_{1}, X_{2} and X_{3} are pairwise independent

(C) (X_{1}, X_{2}) and X_{3} are independently distributed

(D) Variance of X_{1} + X_{2} is π^{2}

33. Suppose that P_{1} and P_{2} are two populations having bivariate normal distributions with mean vectors respectively, and the same variance-covariance matrix two new observations. If the prior probabilities for P_{1} and P_{2} are assumed to be equal and the misclassification costs are also assumed to be equal then, according to linear discriminant rule,

(A) Z_{1} is assigned to P_{1} and Z_{2} is assigned to P_{2}

(B) Z_{1} is assigned to P_{2} and Z_{2} is assigned to P_{1}

(C) both Z_{1} and Z_{2} are assigned to P_{1}

(D) both Z_{1} and Z_{2} are assigned to P_{2}

34. Let X_{1}, …, X_{n} be a random sample of size n (≥2) from an exponential distribution with the probability density function

where θ ∈ (0, ∞). Which of the following statements is TRUE?

35. Let the joint distribution of (X, Y) be bivariate normal with mean vector and variance-covariance matrix where −1 < ρ < 1. Then E[max(X, Y)] equals

(A)

(B)

(C) 0

(D) 1/2

36. Let be independent and identically distributed random vectors, where I_{3} is the 3 × 3 identity matrix. Let

where J_{3} is the 3 × 3 matrix with each entry 1 and for any column vector denotes its transpose. Then the distribution of T is

(A) central chi-square with 5 degrees of freedom

(B) central chi-square with 10 degrees of freedom

(C) central chi-square with 20 degrees of freedom

(D) central chi-square with 30 degrees of freedom

37. Let be independent and identically distributed random vectors, where ∑ is a positive definite matrix. Further, let be a 3 × 4 matrix, where for any matrix M, M^{t} denotes its transpose. If W_{m} (n, ∑) denotes a Wishart distribution of order m with n degrees of freedom and variance-covariance matrix ∑, then which of the following statements is TRUE?

(A) ∑^{−}^{1/2} X^{t}X∑^{−}^{1/2} follows W_{4}(3, I_{4}) distribution

(B) ∑^{−}^{1/2}X^{t}X∑^{−}^{1/2} follows W_{3}(4, I_{3}) distribution

(C) Trace(X∑^{−}^{1}X^{t}) follows χ^{2}_{4} distribution

(D) X^{t}X follows W_{3}(4, ∑) distribution

38. Let the joint distribution of the random variables X_{1}, X_{2} and X_{3} be where

Then which of the following statements is TRUE?

(A) X_{1} – X_{2} + X_{3} and X_{1} are independent

(B) X_{1} + X_{2} and X_{3} – X_{1} are independent

(C) X_{1} – X_{2} + X_{3} and X_{1} + X_{2} are independent

(D) X_{1} – 2X_{2} and 2X_{1} + X_{2} are independent

39. Consider the following one-way fixed effects analysis of variance model

Y_{ij} = μ + τ_{i} + ϵ_{ij}, i = 1, 2, 3; j = 1, 2, 3, 4;

where ϵ_{ij}’s are independent and identically distributed N(0, σ^{2}) random variables, σ ∈ (0, ∞) and τ_{1} + τ_{2} + τ_{3} = 0. Let MST and MSE denote the mean sum of squares due to treatment and the mean sum of squares due to error, respectively. For testing H_{0} : τ_{1} = τ_{2} = τ_{3} = 0 against H_{1} : τ_{i} ≠ 0, for some i = 1, 2, 3, consider the test based on the statistic For positive integers v_{1} and v_{2}, let Fv_{1},v_{2} be a random variable having the central F-distribution with v_{1} and v_{2} degrees of freedom. If the observed value of is given to be 104.45, then the p-value of this test equals

(A) P(F_{2, 9} > 104.45)

(B) P(F_{9, 2 }< 104.45)

(C) P(F_{3, 11} < 104.45)

(D) P(F_{2, 6} > 104.45)

40. Let X_{1}, …, X_{n} be a random sample of size n (≥ 2) from N(θ, 1) distribution, where θ ∈ (−∞, ∞). Consider the problem of testing H_{0} : θ ∈ [1, 2] against H_{1} : θ < 1 or θ > 2, based on X_{1}, …, X_{n}. Which of the following statements is TRUE?

(A) Critical region, of level α (0 < α < 1) of uniformly most powerful test for H_{0} against H_{1} is of the form where c_{1} and c_{2} are such that test is of level α

(B) Critical region, of level α(0 < α < 1) of uniformly most powerful test for H_{0} against H_{1} is of the form {(x_{1}, …, x_{n}) : where c and d are such that the test is level α

(C) At any level α ∈ (0, 1), uniformly most powerful test for H_{0} against H_{1} does NOT exist

(D) At any level α ∈ (0, 1), the power of uniformly most powerful test for H_{0} against H_{1} is less than α

41. In a pure birth process with birth rates λ_{n} = 2^{n}, n ≥ 0, let the random variable T denote the time taken for the population size to grow from 0 to 5. If Var(T) denotes the variance of the random variable T, then 256 × Var(T) = _______

42. Let {X_{n}}_{n}_{≥}_{0} be a homogenous Markov chain whose state space is {0, 1, 2} and whose one-step transition probability matrix is Then _____ (correct up to one decimal place).

43. Let (X, Y) be a random vector such that, for any y > 0, t he conditional probability density function of X given by Y = y is f_{x|Y=y} (x) = ye^{−}^{yx}, x > 0.

If the marginal probability density function of Y is g(y) = ye^{−}^{y}, y > 0 then E(Y|X = 1) = ______ (correct up to one decimal place).

44. Let (X, Y) be a random vector with the joint moment generating function

Let Φ(∙) denote the distribution function of the standard normal distribution and p = P(X + 2Y < 1). If Φ(0) = 0.5, Φ(0.5) = 0.6915, Φ(1) = 0.8413 and Φ(1.5) = 0.9332 then the value of 2p + 1 (round off to two decimal places) equals______

45. Consider a homogeneous Markov chain {X_{n}}_{n}_{≥}_{0} with state space {0, 1, 2, 3} and one-step transition probability matrix

Assume that P(X_{0} = 1) = 1. Let p be the probability that state 0 will be visited before state 3. Then 6 × p = ______

46. Let (X, Y) be a random vector with joint probability mass function

where Then the variance of Y equals ______

47. Let X be a discrete random variable with probability mass function f ∈ {f_{0}, f_{1}}, where

The power of the most powerful level α = 0.1 test for testing H_{0} : X ~ f_{0} against H_{1}:X ~ f_{1}, based on X, equals _______ (correct up to two decimal places).

48. Let be a random vector following distribution, where Then the partial correlation coefficient between X_{2} and X_{3}, with fixed X_{1}, equals _______ (correct up to two decimal places).

49. Let X_{1}, X_{2}, X_{3} and X_{4} be a random sample from a population having probability density function f_{θ}(x) = f(x – θ), −∞ < x < ∞, where θ ∈ (−∞, ∞) and f(−x) = f(x), for all x ∈ (−∞, ∞). For testing H_{0}: θ = 0 against H_{1}: θ < 0, let T^{+} denote the Wilcoxon Signed-rank statistic. Then under H_{0}, 32 × P(T^{+} ≤ 5) ______

50. A simple linear regression model with unknown intercept and unknown slope is fitted to the following data

using the method of ordinary least squares. Then the predicted value of y corresponding to x = 5 is _______

51. Let D = {(x, y, z) ∈ ℝ × ℝ × ℝ : 0 ≤ x, y, z ≤ 1, x + y + z ≤ 12}, where ℝ denotes the set of all real numbers. If then 84 × I = _____

52. Let the random vector (X, Y) have the joint distribution function

Let Var(X) and Var(Y) denote the variances of random variables X and Y, respectively. Then 16 Var(X) + 32 Var(Y) = _______

53. Let {X_{n}}_{n}_{≥}_{1} be a sequence of independent and identically distributed random variables with E(X_{1}) = 0, E(X_{1}^{2}) = 1 and E(X_{1}^{4}) = 3. Further, let

If

where Φ(∙) denotes the cumulative distribution function of the standard normal distribution, then c^{2} = ______(correct up to one decimal place.

54. Let the random vector have the joint probability density function

Then the variance of the random variable X_{1} + X_{2} + X_{3} equals ______ (correct up to one decimal place).

55. Let X_{1}, …, X_{5} be a random sample from a distribution with the probability density function

where θ (−∞, ∞). For testing H_{0} : θ = 0 against H_{1} : θ, let be the sign test statistic, where

Then the size of the test, which rejects H_{0} if and only if equals ______ (correct up to one decimal place).

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