**GATE-2021**

**ST: Statistics**

**GA-General Aptitude**

**Q.1 – Q.5 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).**

1. The current population of a city is 11,02,500. If it has been increasing at the rate of 5% per annum, what was its population 2 years ago?

(A) 9,92,500

(B) 9,95,006

(C) 10,00,000

(D) 12,51,506

2. p and q are positive integers and

(A) 3

(B) 7

(C) 9

(D) 11

3.

The least number of squares that must be added so that the line P-Q becomes the line of symmetry is ________

(A) 4

(B) 3

(C) 6

(D) 7

4. Nostalgia is to anticipation as _______ is to ________

Which one of the following options maintains a similar logical relation in the above sentence?

(A) Present, past

(B) Future, past

(C) Past, future

(D) Future, present

5. Consider the following sentences:

(i) I woke up from sleep.

(ii) I woked up from sleep.

(iii) I was woken up from sleep.

(iv) I was wokened up from sleep.

Which of the above sentences are grammatically CORRECT?

(A) (i) and (ii)

(B) (i) and (iii)

(C) (ii) and (iii)

(D) (i) and (iv)

**Q.6 – Q. 10 Multiple Choice Question (MCQ), carry TWO marks each (for each wrong answer: – 2/3).**

6. Given below are two statements and two conclusions.

Statement 1: All purple are green.

Statement 2: All black are green.

Conclusion I: Some black are purple.

Conclusion II: No black is purple.

Based on the above statements and conclusions, which one of the following options is logically CORRECT?

(A) Only conclusion I is correct.

(B) Only conclusion II is correct.

(C) Either conclusion I or II is correct.

(D) Both conclusion I and II are correct.

7. Computers are ubiquitous. They are used to improve efficiency in almost all fields from agriculture to space exploration. Artificial intelligence (AI) is currently a hot topic. AI enables computers to learn, given enough training data. For humans, sitting in front of a computer for long hours can lead to health issues.

Which of the following can be deduced from the above passage?

(i) Nowadays, computers are present in almost all places.

(ii) Computers cannot be used for solving problems in engineering.

(iii) For humans, there are both positive and negative effects of using computers.

(iv) Artificial intelligence can be done without data.

(A) (ii) and (iii)

(B) (ii) and (iv)

(C) (i), (iii) and (iv)

(D) (i) and (iii)

8. Consider a square sheet of side 1 unit. In the first step, it is cut along the main diagonal to get two triangles. In the next step, one of the cut triangles is revolved about its short edge to form a solid cone. The volume of the resulting cone, in cubic units, is ________

(A) π/3

(B) 2π/3

(C) 3π/2

(D) 3π

9.

The number of minutes spent by two students, X and Y, exercising every day in a given week are shown in the bar chart above.

The number of days in the given week in which one of the students spent a minimum of 10% more than the other student, on a given day, is

(A) 4

(B) 5

(C) 6

(D) 7

10.

Corners are cut from an equilateral triangle to produce a regular convex hexagon as shown in the figure above.

The ratio of the area of the regular convex hexagon to the area of the original equilateral triangle is

(A) 2 : 3

(B) 3 : 4

(C) 4 : 5

(D) 5 : 6

**Statistics (ST)**

**Q.1 – Q.9 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong answer: – 1/3).**

1. Let X be a non-constant positive random variable such that E(X) = 9. Then which one of the following statements is true?

2. Let {W(t)}_{t≥0} be a standard Brownian motion. Then the variance of W(1)W(2) equals

(A) 1

(B) 2

(C) 3

(D) 4

3. Let X_{1},X_{2}, … ,X_{n} be a random sample of size n (≥ 2) from a distributionhaving the probability density function

where θ ∈ (0, ∞). Then the method of moments estimator of θ equals

4. Let {x_{1}, x_{2}, … , xn} be a realization of a random sample of size n (≥ 2) from a N(μ,σ^{2}) distribution, where −∞ < μ < ∞ and σ > 0. Which of the following statements is/are true?

P : 95% confidence interval of μ based on {x_{1}, x_{2}, … , xn} is unique when σ is known.

Q : 95% confidence interval of μ based on {x_{1}, x_{2}, … , x_{n}} is NOT unique when σ is unknown.

(A) P only

(B) Q only

(C) Both P and Q

(D) Neither P nor Q

5. Let X_{1}, X_{2}, … , X_{n} be a random sample of size n (≥ 2) from a N(0, σ_{2}) distribution. For a given σ > 0, let f_{σ} denote the joint probability density function of (X_{1},X_{2}, … ,X_{n}) and S = {f_{σ}: σ > 0}. Let For any positive integer ν and any α ∈ (0, 1), let denote the (1 − α)-th quantile of the central chi-square distribution with ν degrees of freedom. Consider testing H_{0}: σ = 1 against H_{1}: σ > 1 at level α. Then which one of the following statements is true?

(A) S has a monotone likelihood ratio in T_{1} and H_{0} is rejected if

(B) S has a monotone likelihood ratio in T_{1} and H_{0} is rejected if

(C) S has a monotone likelihood ratio in T_{2} and H_{0} is rejected if

(D) S has a monotone likelihood ratio in T_{2} and H_{0} is rejected if

6. Let X and Y be two random variables such that p_{11} + p_{10} + p_{01} + p_{00} = 1, where p_{ij} = P(X = i, Y = j), i,j = 0, 1. Suppose that a realization of a random sample of size 60 from the joint distribution of (X, Y) gives

n_{11} = 10, n_{10} = 20, n_{01} = 20 and n_{00} = 10,

where nij denotes the frequency of (i, j) for i, j = 0, 1. If the chi-square test of independence is used to test

H_{0}: p_{ij} = p_{i}⋅p_{⋅}_{j} for i, j = 0, 1 against H_{1}: p_{ij} ≠ p_{i}⋅p_{⋅}_{j} for at least one pair

where p_{i}_{⋅} = p_{i0} + p_{i1} and p_{⋅}_{j} = p_{0j} + p_{1j}, then which one of the following statements is true?

(i,j),

(A) Under H_{0}, the test statistic follows central chi-square distribution with one degree of freedom and the observed value of the test statistic is 20/3

(B) Under H_{0}, the test statistic follows central chi-square distribution with three degrees of freedom and the observed value of the test statistic is 20/3

(C) Under H_{0}, the test statistic follows central chi-square distribution with one degree of freedom and the observed value of the test statistic is 16/3

(D) Under H_{0}, the test statistic follows central chi-square distribution with three degrees of freedom and the observed value of the test statistic is 16/3

7. Let the joint distribution of (X, Y) be bivariate normal with mean vector and variance-covariance matrix where −1 < ρ < 1. Let Φ_{ρ}(0, 0) = P(X ≤ 0, Y ≤ 0). Then the Kendall’s τ coefficient between X and Y equals

(A) 4Φ_{ρ}(0,0) − 1

(B) 4Φ_{ρ}(0,0)

(C) 4Φ_{ρ}(0,0) + 1

(D) Φ_{ρ}(0,0)

8. Consider the simple linear regression model

Y_{i} = β_{0} + β_{1}x_{i} + ε_{i}, i = 1, 2, … , n (n ≥ 3),

where β_{0} and β_{1} are unknown parameters and εi’s are independent and identically distributed random variables with mean zero and finite variance σ^{2} > 0. Suppose that are the ordinary least squares estimators of β_{0} and β_{1}, respectively. Define and , where y_{i} 0 is the observed value of Y_{i}, , i = 1, 2, … , n. Then for a real constant c, the variance of is

9. Let __X _{1}__,

__X__,

_{2}__X__,

_{3}__Y__,

_{1}__Y__,

_{2}__Y__and

_{3}__Y__be independent random vectors such that

_{4}__X__follows N4(

_{i}__0__, Σ

_{1}) distribution for i = 1, 2, 3, and

__Y__follows N4(

_{j}__0__, Σ

_{2}) distribution for j = 1, 2, 3, 4, where Σ

_{1}and Σ

_{2}are positive definite matrices. Further, let

where X = [__X _{1}__

__X__

_{2}__X__] is a 4 × 3 matrix, Y = [Y1 Y2 Y3 Y4] is a 4 × 4 matrix and X

_{3}^{T}and Y

^{T}denote transposes of X and Y, respectively. If W

_{m}(n, Σ) denotes a Wishart distribution of order m with n degrees of freedom and variance-covariance matrix Σ and I

_{n}denotes the n × n identity matrix, then which one of the following statements is true?

(A) Z follows W_{4}(7,I4) distribution

(B) Z follows W_{4}(4,I4) distribution

(C) Z follows W_{7} (4,I7 ) distribution

(D) Z follows W_{7} (7,I7) distribution

**Q.10 – Q.25 Numerical Answer Type (NAT), carry ONE mark each (no negative marks).**

10.

11. Let

Then the value of e I^{1+ π }equals __________ (round off to 2 decimal places).

12. Let and I3 be the 3 × 3 identity matrix. Then the nullity of 5A(I3 + A ^{2}) equals __________

13. Let A be the 2 × 2 real matrix having eigenvalues 1 and − 1, with corresponding eigenvectors respectively. If then a + b + c + d equals __________ (round off to 2 decimal places).

14. Let A and B be two events such that P(B) = 3/4 and P(A ⋃ B^{C}) = 1/2. If A and B are independent, then P(A) equals __________ (round off to 2 decimal places).

15. A fair die is rolled twice independently. Let X and Y denote the outcomes of the first and second roll, respectively. Then E(X + Y | (X − Y)^{2} = 1) equals ________

16. Let X be a random variable having distribution function

where a and c are appropriate constants. Let n ≥ 1, and If P(X ≤ 1) = 1/2 and E(X) = 5/3, then P(X ∈ A) equals __________ (round off to 2 decimal places).

17. If the marginal probability density function of the k^{th} order statistic of a random sample of size 8 from a uniform distribution on [0, 2] is then k equals __________

18. For α > 0, let be a sequence of independent random variables such that

Let S = {α > 0 ∶ X_{n}^{(a)} converges to 0 almost surely as n → ∞}. Then the infimum of S equals __________ (round off to 2 decimal places).

19. Let {Xn}_{n≥1} be a sequence of independent and identically distributed random variables each having uniform distribution on [0, 2]. For n ≥ 1, let

Then, as n → ∞, the sequence {Z_{n}}_{n≥1} converges almost surely to _______ (round off to 2 decimal places).

20. Let {Xn}_{n≥0} be a time-homogeneous discrete time Markov chain with state space {0, 1} and transition probability matrix If P(X_{0} = 0) = P(X_{0} = 1) = 0. 5, then equals ______

21. Let {0, 2} be a realization of a random sample of size 2 from a binomial distribution with parameters 2 and p, where p ∈ (0, 1). To test H0: p = 1/2 against H_{1}: p ≠ 1 the observed value of the likelihood ratio test statistic equals _________ (round off to 2 decimal places).

22. Let X be a random variable having the probability density function

Then equals __________ (round off to 2 decimal places).

23. Let (Y, X_{1},X_{2}) be a random vector with mean vector and variance-covariance matrix Then the value of the multiple correlation coefficient between Y and its best linear predictor on X_{1} and X_{2} equals _________ (round off to 2 decimal places).

24. Let be a random sample from a bivariate normal distribution with unknown mean vector __μ__ and unknown variance-covariance matrix Σ, which is a positive definite matrix. The p-value corresponding to the likelihood ratio test for testing based on the realization of the random sample equals __________ (round off to 2 decimal places).

25. Let Y_{i} = α + βx_{i} + ε_{i}, i = 1, 2, 3, where x_{i}’s are fixed covariates, α and β are unknown parameters and ε_{i}’s are independent and identically distributed random variables with mean zero and finite variance. Let be the ordinary least squares estimators of α and β, respectively. Given the following observations

the value of equals ________ (round off to 2 decimal places).

**Q.26 – Q.43 Multiple Choice Question (MCQ), carry TWO mark each (for each wrong answer: – 2/3).**

26. Let f: ℝ → ℝ be defined by

where ℝ denotes the set of all real numbers, ℤ denotes the set of all integers, ℕ denotes the set of all positive integers and gcd(p, q) denotes the greatest common divisor of p and q. Then which one of the following statements is true?

(A) f is not continuous at 0

(B) f is not differentiable at 0

(C) f is differentiable at 0 and the derivative of f at 0 equals 0

(D) f is differentiable at 0 and the derivative of f at 0 equals 1

27. Let f:[0, ∞) → ℝ be a function, where R denotes the set of all real numbers. Then which one of the following statements is true?

(A) If f is bounded and continuous, then f is uniformly continuous

(B) If f is uniformly continuous, then exists

(C) If f is uniformly continuous, then the function g(x) = f(x) sin x is also uniformly continuous

(D) If f is continuous and is finite, then f is uniformly continuous

28. Let f: ℝ → ℝ be a differentiable function such that f(0) = 0 and f′ (x) + 2f(x) > 0 for all x ∈ R, where f′ denotes the derivative of f and R denotes the set of all real numbers. Then which one of the following statements is true?

(A) f(x) > 0, for all x > 0 and f(x) < 0, for all x < 0

(B) f(x) < 0, for all x ≠ 0

(C) f(x) > 0, for all x ≠ 0

(D) f(x) < 0, for all x > 0 and f(x) > 0, for all x < 0

29. Let M be the collection of all 3 × 3 real symmetric positive definite matrices. Consider the set where 0 denotes the 3 × 3 zero matrix. Then the number of elements in S equals

(A) 0

(B) 1

(C) 8

(D) ∞

30. Let A be a 3 × 3 real matrix such that I_{3} + A is invertible and let B = (I_{3} + A)^{−}^{1}(I_{3} – A), where I_{3} denotes the 3 × 3 identity matrix. Then which one of the following statements is true?

(A) If B is orthogonal, then A is invertible

(B) If B is orthogonal, then all the eigenvalues of A are real

(C) If B is skew-symmetric, then A is orthogonal

(D) If B is skew-symmetric, then the determinant of A equals −1

31. Let X be a random variable having Poisson distribution such thatE(X^{2}) = 110. Then which one of the following statements is NOT true?

(A) E(X^{n}) = 10 E[(X + 1)n^{−1}], for all n = 1, 2, 3, …

(B)

(C) P(X = k) < P(X = k + 1), for k = 0, 1, … , 8

(D) P(X = k) > P(X = k + 1), for k = 10, 11, …

32. Let X be a random variable having uniform distribution on Then which one of the following statements is NOT true?

(A) Y = cot X follows standard Cauchy distribution

(B) Y = tan X follows standard Cauchy distribution

(C) has moment generating function

(D) follows central chi-square distribution with one degree of freedom

33. Let Ω = {1, 2, 3, … } represent the collection of all possible outcomes of a random experiment with probabilities P({n}) = αn for n ∈ Ω. Then which one of the following statements is NOT true?

(A)

(B)

(C) For any positive integer k, there exist k disjoint events A_{1}, A_{2}, … , A_{k} such that

(D) There exists a sequence {Ai}_{i≥1} of strictly increasing events such that

34. Let (X, Y) have the joint probability density function

Then which one of the following statements is NOT true?

(A) The probability density function of X + Y is

(B) P(X + Y > 4) = 3/4

(C) E(X + Y) = 4 log_{e} 2

(D) E(Y | X = 2) = 4

35. Let X_{1}, X_{2} and X_{3} be three uncorrelated random variables with common variance σ^{2} < ∞. Let Y_{1} = 2X_{1} + X_{2} + X_{3}, Y_{2} = X_{1} + 2X_{2} + X_{3} and Y_{3} = X_{1} + X_{2} + 2X_{3}. Then which of the following statements is/are true?

P : The sum of eigenvalues of the variance covariance matrix of (Y_{1}, Y_{2}, Y_{3}) is 18σ^{2}.

Q : The correlation coefficient between Y_{1} and Y_{2} equals that between Y_{2} and Y_{3}.

(A) P only

(B) Q only

(C) Both P and Q

(D) Neither P nor Q

36. Let {Xn}_{n≥0} be a time-homogeneous discrete time Markov chain with either finite or countable state space S. Then which one of the following statements is true?

(A) There is at least one recurrent state

(B) If there is an absorbing state, then there exists at least one stationary distribution

(C) If all the states are positive recurrent, then there exists a unique stationary distribution

(D) If {Xn}_{n≥0} is irreducible, S = {1, 2} and [π_{1} π_{2}] is a stationary distribution, then for i = 1, 2

37. Let customers arrive at a departmental store according to a Poisson process with rate 10. Further, suppose that each arriving customer is either a male or a female with probability 1/2 each, independent of all other arrivals. Let N(t) denote the total number of customers who have arrived by time t. Then which one of the following statements is NOT true?

(A) If S_{2} denotes the time of arrival of the second female customer, then

(B) If M(t) denotes the number of male customers who have arrived by time t, then

(C) E [(N(t))^{2}] = 100t^{2} + 10t

(D) E[N(t)N(2t)] = 200t^{2} + 10t

38. Let X_{(1)} < X_{(2)} < X_{(3)} < X_{(4)} < X_{(5)}be the order statistics corresponding to a random sample of size 5 from a uniform distribution on [0, θ], where θ ∈ (0, ∞). Then which of the following statements is/are true?

P : 3X_{(2)} is an unbiased estimator of θ.

Q : The variance of E[2X_{(3)} | X_{(5)}] is less than or equal to the variance of 2X_{(3)}.

(A) P only

(B) Q only

(C) Both P and Q

(D) Neither P nor Q

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

where θ ∈ (0, ∞). Let X_{(1)} = min{ X_{1}, X_{2}, … , X_{n}} and Then E(X_{(1)}| T ) equals

(A) T/n^{2}

(B) T/n

(C)

(D)

40. Let X_{1}, X_{2}, … , X_{n} be a random sample of size n (≥ 2) from a uniform distribution on [−θ, θ], where θ ∈ (0, ∞). Let X_{(1)} = min{X_{1}, X_{2}, … , X_{n}} and X_{(n)} = max{X_{1}, X_{2}, … ,X_{n}}. Then which of the following statements is/are true?

P : (X_{(1)}, X_{(n)}) is a complete statistic.

Q : X_{(n)} − X_{(1)} is an ancillary statistic.

(A) P only

(B) Q only

(C) Both P and Q

(D) Neither P nor Q

41. Let {X_{n}}_{n≥1} be a sequence of independent and identically distributed random variables having common distribution function F(⋅). Let a < b be two real numbers such that F(x) = 0 for all x ≤ a, 0 < F(x) < 1 for all a < x < b and F(x) = 1 for all x ≥ b. Let S_{n}(x) be the empirical distribution function at x based on X_{1}, X_{2}, … ,X_{n}, n ≥ 1. Then which one of the following statements is NOT true?

(A)

(B) For fixed x ∈ (a, b) and t ∈ (− ∞, ∞), where Z is the standard normal random variable

(C) The covariance between S_{n}(x) and S_{n}(y) equals for all n ≥ 2 and for fixed −∞ < x, y < ∞

(D) If then {4n Y_{n}}_{n≥1} converges in distribution to a central chi-square random variable with 2 degrees of freedom

42. Let the joint distribution of random variables X_{1},X_{2},X_{3} and X_{4} be where

Then which one of the following statements is true?

(A) follows a central chi-square distribution with 2 degrees of freedom

(B) follows a central chi-square distribution with 2 degrees of freedom

(C) is NOT finite

(D) is NOT finite

43. Let distribution, where I_{8} is the 8 × 8 identity matrix. Let be independent and follow central chi-square distributions with 3 and 4 degrees of freedom, respectively, where Σ_{1} and Σ_{2} are 8 × 8 matrices and denotes transpose of Then which of the following statements is/are true?

P : Σ_{1} and Σ_{2} are idempotent.

Q : Σ_{1}Σ_{2} = 0, where 0 is the 8 × 8 zero matrix.

(A) P only

(B) Q only

(C) Both P and Q

(D) Neither P nor Q

**Q.44 – Q.55 Numerical Answer Type (NAT), carry TWO mark each (no negative marks).**

44. Let (X, Y) have a bivariate normal distribution with the joint probability density function

Then 8 E(XY) equals ________

45. Let f: ℝ × ℝ → ℝ be defined by f(x, y) = 8x^{2} − 2y, where ℝ denotes the set of all real numbers. If M and m denote the maximum and minimum values of f, respectively, on the set {(x, y) ∈ ℝ × ℝ ∶ x^{2} + y^{2} = 1}, then M − m equals ________ (round off to 2 decimal places).

46. Let A = [a u_{1} u_{2} u_{3}], B = [b u_{1} u_{2} u_{3}] and C = [u_{2} u_{3} u_{1} a + b] be three 4 × 4 real matrices, where a, b, u_{1}, u_{2} and u_{3} are 4 × 1 real column vectors. Let det(A), det(B) and det(C) denote the determinants of the matrices A, B and C, respectively. If det(A) = 6 and det(B) = 2, then det(A + B) − det(C) equals _________

47. Let X be a random variable having the moment generating function

Then P(X > 1) equals __________ (round off to 2 decimal places).

48. Let {X_{n}}_{n≥1} be a sequence of independent and identically distributed random variables each having uniform distribution on [0, 3]. Let Y be a random variable, independent of {X_{n}}_{n≥1}, having probability mass function

Then P(max{X_{1},X_{2}, … ,X_{Y}} ≤ 1) equals __________ (round off to 2 decimal places).

49. Let {X_{n}}_{n≥1} be a sequence of independent and identically distributed random variables each having probability density function

Let X_{(n)} = max{X_{1},X_{2}, … ,X_{n}} for n ≥ 1. If Z is the random variable to which {X_{(n)} − log_{e} n}_{n≥1}converges in distribution, as n → ∞, then the median of Z equals __________ (round off to 2 decimal places).

50. Consider an amusement park where visitors are arriving according to a Poisson process with rate 1. Upon arrival, a visitor spends a random amount of time in the park and then departs. The time spent by the visitors are independent of one another, as well as of the arrival process, and have common probability density function

If at a given time point, there are 10 visitors in the park and p is the probability that there will be exactly two more arrivals before the next departure, then 1/p equals ________

51. Let {0. 90, 0. 50, 0. 01, 0. 95} be a realization of a random sample of size 4 from the probability density function

where 0. 5 ≤ θ < 1. Then the maximum likelihood estimate of θ based on the observed sample equals __________ (round off to 2 decimal places).

52. Let a random sample of size 100 from a normal population with unknown mean μ and variance 9 give the sample mean 5. 608. Let Φ(⋅) denote the distribution function of the standard normal random variable. If Φ(1. 96) = 0. 975, Φ(1. 64) = 0. 95 and the uniformly most powerful unbiased test based on sample mean is used to test H_{0}: μ = 5. 02 against H_{1}: μ ≠ 5. 02, then the p-value equals __________ (round off to 3 decimal places).

53. Let X be a discrete random variable with probability mass function p ∈ {p_{0}, p_{1}}, where

To test H_{0}: p = p_{0} against H1: p = p_{1}, the power of the most powerful test of size 0. 05, based on X, equals __________ (round off to 2 decimal places).

54. Let X_{1}, X_{2}, … , X_{10} be a random sample from a probability density function f_{θ}(x) = f(x − θ), −∞ < x < ∞, where −∞ < θ < ∞ and f(−x) = f(x) for −∞ < x < ∞. For testing H_{0}: θ = 1. 2 against H_{1}: θ ≠ 1. 2, let T^{+} denote the Wilcoxson Signed-rank test statistic. If η denotes the probability of the event {T^{+} < 50} under H_{0}, then 32 η equals __________ (round off to 2 decimal places).

55. Consider the multiple linear regression model Y_{i} = β_{0} + β_{1}x_{1},i + β_{2}x_{2,i} + ⋯ + β_{22}x_{22,i} + ϵ_{i}, i = 1, 2, … , 123, where, for j = 0, 1, 2, … , 22, β_{j}’s are unknown parameters and ϵ_{i}’s are independent and identically distributed N(0, σ^{2}), σ > 0, random variables.

If the sum of squares due to regression is 338. 92, the total sum of squares is 522. 30 and R^{2}_{adj} denotes the value of adjusted R^{2} then 100 R^{2}_{adj}, equals __________ (round off to 2 decimal places).

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