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) n2 – n
(B) n2 + n
(C) 2n2 – n
(D) 2n2 + 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 = [mij], i. j= 1, 2, 3, 4, the diagonal elements are all zero and mij = −mij. 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 I3 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) I3 – M is idempotent
(D) (I3 + M)−1 = I3 – 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: x2 – y2 = 4} and f : S → ℝ be defined by f(x, y) = 6x + y2, 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
Yijk = μ + αi + βj + ϵijk, i = 1, 2; j = 1, 2, 3; k = 1, 2, 3;
where ϵijk’s are independently and identically distributed N(0, σ2) random variables, σ ∈ (0, ∞), α1 + α2 = 0 and β1 + β2 + β3 = 0. Let SSE denote the sum of squares due to error. For any positive integer v and any α ∈ (0, 1), let χ2v,α denote the (1 – α)-th quantile of the central chi-square distribution with v degrees of freedom. Then a 95% confidence interval for σ2 is given by
7. Let X1, …, X20 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 X1, …, X10 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 χ2v,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 X1, …, Xn 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 X1 …, Xn be a random sample of size n(≥2) from N(θ, 2θ2) distribution, where θ ∈ (0, ∞). Which of the following statements is TRUE?
(A) is the unique unbiased estimator of θ2 that is a function of minimal sufficient statistic
(B) is an unbiased estimator of θ2
(C) There exist infinite number of unbiased estimators of θ2 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
Yi = β0 + β1xi2 + ϵi, i = 1, 2, …, n(n ≥ 2);
where β0 and β1 are unknown parameters and ϵi’s are random errors. Let yi be the observed value of Yi, i = 1, …, n. Using the method of ordinary least squares, the estimate of β1 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) χ2p, the central chi-square distribution with p degrees of freedom
(B) Fp,n−p, the central F distribution with p and n – p degrees of freedom
(C) where Fp,n−p, is the central F distribution with p and n – p degrees of freedom
(D) where Fp,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 X1, …, Xn 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 H0 : θ = 1 against H1 : θ = 2, based on X1, …, Xn. 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 H0 against H1 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(X1 + X2 + X3 + X4 > 0) = _______ (correct up to one decimal place).
18. Let {Xn}n≥0 be a homogeneous Markov chain with state space {0, 1} and one-step transition probability matrix If P(X0 = 0) = 1/3, then 27 × E(X2) = ______ (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 X1, …, Xn be a random sample of size n (≥ 2) from an exponential distribution with the probability density function
where θ ∈ (0, ∞). If X(1) = min{X1, …, Xn} then the conditional expectation
24. Let Yi = α + βxi + ϵi, i = 1, 2, …, 7, where xi’s are fixed covariates and ϵi’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 yi is the observed value of Yi, 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) = x4 – 2x3y + 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 : ℂ3 = 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) T2 = I3, where I3 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) X1, X2 and X3 are mutually independent
(B) X1, X2 and X3 are pairwise independent
(C) (X1, X2) and X3 are independently distributed
(D) Variance of X1 + X2 is π2
33. Suppose that P1 and P2 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 P1 and P2 are assumed to be equal and the misclassification costs are also assumed to be equal then, according to linear discriminant rule,
(A) Z1 is assigned to P1 and Z2 is assigned to P2
(B) Z1 is assigned to P2 and Z2 is assigned to P1
(C) both Z1 and Z2 are assigned to P1
(D) both Z1 and Z2 are assigned to P2
34. Let X1, …, Xn 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 I3 is the 3 × 3 identity matrix. Let
where J3 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, Mt denotes its transpose. If Wm (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 XtX∑−1/2 follows W4(3, I4) distribution
(B) ∑−1/2XtX∑−1/2 follows W3(4, I3) distribution
(C) Trace(X∑−1Xt) follows χ24 distribution
(D) XtX follows W3(4, ∑) distribution
38. Let the joint distribution of the random variables X1, X2 and X3 be where
Then which of the following statements is TRUE?
(A) X1 – X2 + X3 and X1 are independent
(B) X1 + X2 and X3 – X1 are independent
(C) X1 – X2 + X3 and X1 + X2 are independent
(D) X1 – 2X2 and 2X1 + X2 are independent
39. Consider the following one-way fixed effects analysis of variance model
Yij = μ + τi + ϵij, i = 1, 2, 3; j = 1, 2, 3, 4;
where ϵij’s are independent and identically distributed N(0, σ2) random variables, σ ∈ (0, ∞) and τ1 + τ2 + τ3 = 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 H0 : τ1 = τ2 = τ3 = 0 against H1 : τi ≠ 0, for some i = 1, 2, 3, consider the test based on the statistic For positive integers v1 and v2, let Fv1,v2 be a random variable having the central F-distribution with v1 and v2 degrees of freedom. If the observed value of
is given to be 104.45, then the p-value of this test equals
(A) P(F2, 9 > 104.45)
(B) P(F9, 2 < 104.45)
(C) P(F3, 11 < 104.45)
(D) P(F2, 6 > 104.45)
40. Let X1, …, Xn be a random sample of size n (≥ 2) from N(θ, 1) distribution, where θ ∈ (−∞, ∞). Consider the problem of testing H0 : θ ∈ [1, 2] against H1 : θ < 1 or θ > 2, based on X1, …, Xn. Which of the following statements is TRUE?
(A) Critical region, of level α (0 < α < 1) of uniformly most powerful test for H0 against H1 is of the form where c1 and c2 are such that test is of level α
(B) Critical region, of level α(0 < α < 1) of uniformly most powerful test for H0 against H1 is of the form {(x1, …, xn) : where c and d are such that the test is level α
(C) At any level α ∈ (0, 1), uniformly most powerful test for H0 against H1 does NOT exist
(D) At any level α ∈ (0, 1), the power of uniformly most powerful test for H0 against H1 is less than α
41. In a pure birth process with birth rates λn = 2n, 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 {Xn}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 fx|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 {Xn}n≥0 with state space {0, 1, 2, 3} and one-step transition probability matrix
Assume that P(X0 = 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 ∈ {f0, f1}, where
The power of the most powerful level α = 0.1 test for testing H0 : X ~ f0 against H1:X ~ f1, 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 X2 and X3, with fixed X1, equals _______ (correct up to two decimal places).
49. Let X1, X2, X3 and X4 be a random sample from a population having probability density function fθ(x) = f(x – θ), −∞ < x < ∞, where θ ∈ (−∞, ∞) and f(−x) = f(x), for all x ∈ (−∞, ∞). For testing H0: θ = 0 against H1: θ < 0, let T+ denote the Wilcoxon Signed-rank statistic. Then under H0, 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 {Xn}n≥1 be a sequence of independent and identically distributed random variables with E(X1) = 0, E(X12) = 1 and E(X14) = 3. Further, let
If
where Φ(∙) denotes the cumulative distribution function of the standard normal distribution, then c2 = ______(correct up to one decimal place.
54. Let the random vector have the joint probability density function
Then the variance of the random variable X1 + X2 + X3 equals ______ (correct up to one decimal place).
55. Let X1, …, X5 be a random sample from a distribution with the probability density function
where θ (−∞, ∞). For testing H0 : θ = 0 against H1 : θ, let be the sign test statistic, where
Then the size of the test, which rejects H0 if and only if equals ______ (correct up to one decimal place).
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