**GATE-2022**

**ST: Statistics**

**General Aptitude**

**Q.1 – Q.5 Carry ONE mark each.**

1. X speaks _________ Japanese _________ Chinese.

(A) neither / or

(B) either / nor

(C) neither / nor

(D) also / but

2. A sum of money is to be distributed among P, Q, R, and S in the proportion 5 : 2 : 4 : 3, respectively.

If R gets Rs 1000 more than S, what is the share of Q (in Rs)?

(A) 500

(B) 1000

(C) 1500

(D) 2000

3. A trapezium has vertices marked as P, Q, R and S (in that order anticlockwise). The side PQ is parallel to side SR.

Further, it is given that, PQ = 11 cm, QR = 4 cm, RS = 6 cm and SP = 3 cm.

What is the shortest distance between PQ and SR (in cm)?

(A) 1.80

(B) 2.40

(C) 4.20

(D) 5.76

4. The figure shows a grid formed by a collection of unit squares. The unshaded unit square in the grid represents a hole.

What is the maximum number of squares without a “hole in the interior” that can be formed within the 4 × 4 grid using the unit squares as building blocks?

(A) 15

(B) 20

(C) 21

(D) 26

5. An art gallery engages a security guard to ensure that the items displayed are protected. The diagram below represents the plan of the gallery where the boundary walls are opaque. The location the security guard posted is identified such that all the inner space (shaded region in the plan) of the gallery is within the line of sight of the security guard.

If the security guard does not move around the posted location and has a 360o view, which one of the following correctly represents the set of ALL possible locations among the locations P, Q, R and S, where the security guard can be posted to watch over the entire inner space of the gallery.

(A) P and Q

(B) Q

(C) Q and S

(D) R and S

**Q.6 – Q.10 Carry TWO marks each.**

6. Mosquitoes pose a threat to human health. Controlling mosquitoes using chemicals may have undesired consequences. In Florida, authorities have used genetically modified mosquitoes to control the overall mosquito population. It remains to be seen if this novel approach has unforeseen consequences.

Which one of the following is the correct logical inference based on the information in the above passage?

(A) Using chemicals to kill mosquitoes is better than using genetically modified mosquitoes because genetic engineering is dangerous

(B) Using genetically modified mosquitoes is better than using chemicals to kill mosquitoes because they do not have any side effects

(C) Both using genetically modified mosquitoes and chemicals have undesired consequences and can be dangerous

(D) Using chemicals to kill mosquitoes may have undesired consequences but it isnot clear if using genetically modified mosquitoes has any negative consequence

7. Consider the following inequalities.

(i) 2x − 1 > 7

(ii) 2x − 9 < 1

Which one of the following expressions below satisfies the above two inequalities?

(A) x ≤ −4

(B) −4 <x ≤ 4

(C) 4 <x< 5

(D) x ≥ 5

8. Four points P(0, 1), Q(0, −3), R(−2, −1), and S(2, −1) represent the vertices of a quadrilateral.

What is the area enclosed by the quadrilateral?

(A) 4

(B) 4√2

(C) 8

(D) 8√2

9. In a class of five students P, Q, R, S and T, only one student is known to have copied in the exam. The disciplinary committee has investigated the situation and recorded the statements from the students as given below.

Statement of P: R has copied in the exam.

Statement of Q: S has copied in the exam.

Statement of R: P did not copy in the exam.

Statement of S: Only one of us is telling the truth.

Statement of T: R is telling the truth.

The investigating team had authentic information that S never lies.

Based on the information given above, the person who has copied in the exam is

(A) R

(B) P

(C) Q

(D) T

10. Consider the following square with the four corners and the center marked as P, Q, R, S and T respectively.

Let X, Y and Z represent the following operations:

X: rotation of the square by 180 degree with respect to the

S-Q axis.

Y: rotation of the square by 180 degree with respect to the P-R axis.

Z: rotation of the square by 90 degree clockwise with respect to the axis perpendicular, going into the screen and passing through the point T.

Consider the following three distinct sequences of operation (which are applied

in the left to right order).

(1) XYZZ

(2) XY

(3) ZZZZ

Which one of the following statements is correct as per the information provided above?

(A) The sequence of operations (1) and (2) are equivalent

(B) The sequence of operations (1) and (3) are equivalent

(C) The sequence of operations (2) and (3) are equivalent

(D) The sequence of operations (1), (2) and (3) are equivalent

**Statistics**

**Q.11 – Q.35 Carry ONE mark Each**

11. Let M be a 2 × 2 real matrix such that (I + M)^{−}^{1} = I – αM, where α is a non-zero real number and I is the 2 × 2 identity matrix. If the trace of the matrix M is 3, then the value of 𝛼 is

(A) 3/4

(B) 1/3

(C) 1/2

(D) 1/4

12. Let {X(t)}_{t}_{≥}_{0} be a linear pure death process with death rate μ_{i} = 5i,, i = 0.1, …, N, N ≥ Suppose that p_{i}(t) = P(X(t) = i). Then the system of forward Kolmogorov’s equations is

13. Let S^{2} be the variance of a random sample of size n > 1 from a normal population with an unknown mean μ and an unknown finite variance σ^{2} > 0. Consider the following statements:

(I) S^{2} is an unbiased estimator of σ^{2}, and S is an unbiased estimator of σ.

(II) is a maximum likelihood estimator of σ^{2}, and is a maximum likelihood estimator of σ.

Which of the above statements is/are true?

(A) (I) only

(B) (II) only

(C) Both (I) and (II)

(D) Neither (I) nor (II)

14. Let f : ℝ^{2} → ℝ be a function defined by

Then which one of the following statements is true?

(A) f is bounded and ∂f/∂x is unbounded on ℝ^{2}

(B) f is unbounded and ∂f/∂x is bounded on ℝ^{2}

(C) Both f and ∂f/∂x are unbounded on ℝ^{2}

(D) Both f and ∂f/∂x are bounded on ℝ^{2}

15. Let X_{1}, X_{2}, … , X_{n} be a random sample from a distribution with cumulative distribution function F(x). Let the empirical distribution function of the sample be F_{n}(x). The classical Kolmogorov-Smirnov goodness of fit test statistic is given by

Consider the following statements:

(I) The distribution of T_{n} is the same for all continuous underlying

distribution functions F(x).

(II) D_{n} converges to 0 almost surely, as n → ∞.

Which of the above statements is/are true?

(A) (I) only

(B) (II) only

(C) Both (I) and (II)

(D) Neither (I) nor (II)

16. Consider the following transition matrices P_{1} and P_{2} of two Markov chains:

Then which one of the following statements is true?

(A) Both P_{1} and P_{2} have unique stationary distributions

(B) P_{1} has a unique stationary distribution, but P_{2} has infinitely many stationary distributions

(C) P_{1} has infinitely many stationary distributions, but P_{2} has a unique stationary distribution

(D) Neither P_{1} nor P_{2} has unique stationary distribution

17. Let X_{1}, X_{2}, …, X_{20} be a random sample of size 20 from N_{6}(μ, ∑), with det(∑) ≠ 0, and suppose both μ and ∑ are unknown. Let

Consider the following two statements:

(I) The distribution of 19 S is W_{6}(19, ∑) (Wishart distribution of order 6 with 19 degrees of freedom).

(II) The distribution of (X_{3} − μ)^{T} S^{−1}(X_{3} − μ) is χ_{6}^{2} (Chi-square distribution with 6 degrees of freedom).

Then which of the above statements is/are true?

(A) (I) only

(B) (II) only

(C) Both (I) and (II)

(D) Neither (I) nor (II)

18. Let X_{1}, X_{2}, …, X_{3} be a random sample from the distribution

Let χ^{2} _{α}_{,n} denote the value of a Chi-square random variable Y with n degrees of freedom such that P(Y > χ^{2} _{α}_{,n} = α. If x_{1}, x_{2}, …, x_{18} is a realization of this random sample, then, based on the sufficient statistic which one of the following is a 98% confidence interval for θ?

19. Let X_{1}, X_{2}, …, X_{n} be a random sample from a population f (x; θ), where θ is a parameter. Then which one of the following statements is NOT true?

(A) is a complete and sufficient statistic for θ, if x = 0, 1, 2, …, and θ > 0

(B) is a complete and sufficient statistic for θ, if −∞ < x < ∞, θ > 0

(C) f(x, θ) = θx^{θ−}^{1}, 0 x < 1, θ > 0 has monotone likelihood ratio property in

(D) X_{(n)} – X_{(1)} is ancillary statistic for θ if f(x; θ) = 1, 0 < θ < x < θ +1, where X_{(1)} = min{X_{1}, X_{2}, …, X_{n}) and X_{(n)} = max[X_{1}, X_{2}, …, X_{n}}

20. A random sample X_{1}, X_{2}, …, X_{6} of size 6 is taken from a Bernoulli distribution with the parameter θ. The null hypothesis H_{0} : θ = 1/2 is to be tested against the alternative hypothesis H_{1} : θ > 1/2, based on the statistic If the value of Y corresponding to the observed sample values is 4, then the 𝑝-value of the test statistic is

(A) 21/32

(B) 9/64

(C) 11/32

(D) 7/64

21. Let be a sequence of positive real numbers satisfying with a_{1} = 3 and a_{n} < 7 for all n ≥

Consider the following statements:

(I) {a_{n}} is monotonically increasing.

(II) {a_{n}} converges to a value in the interval [3, 7]

Then which of the above statements is/are true?

(A) (I) only

(B) (II) only

(C) Both (I) and (II)

(D) Neither (I) nor (II)

22. Let M be any square matrix of arbitrary order 𝑛 such that M^{2} = 0 and the nullity of M is 6. Then the maximum possible value of n (in integer) is ______

23. Consider the usual inner product in ℝ^{4}. Let u ∈ ℝ^{4} be a unit vector orthogonal to the subspace

S = {(x_{1}, x_{2}, x_{3}, x_{4})^{T} ∈ ℝ^{4}|x_{1} + x_{2} + x_{3} + x_{4} = 0}.

If v = (1, −2, 1, 1)^{T}, and the vectors u and v – αu, α ∈ ℝ, are orthogonal, then the value of α^{2} (rounded off to two decimal places) is equal to _______

24. Let {B(t)}_{t}_{≥}_{0} be a standard Brownian motion and let ϕ (∙) be the cumulative distribution function of the standard normal distribution. If α > 0, then the value of α (in integer) is equal to ____

25. Let X and Y be two independent exponential random variables with E(X^{2}) = 1/2 and E(Y^{2}) = 2/9. Then P(X < 2Y) (rounded off to two decimal places) is equal to ______

26. Let X be a random variable with the probability mass function x = 1, 2, 3 …. . Then the value of (rounded off to two decimal places) is equal to _______

27. Let X_{i}, i = 1, 2, …, n, be i.i.d. random variables from a normal distribution with mean 1 and variance 4 Let If Var(S_{n}) denotes the variance of S_{n}, then the value of (in integer) is equal to ______

28. At a telephone exchange, telephone calls arrive independently at an average rate of 1 call per minute, and the number of telephone calls follows a Poisson distribution. Five time intervals, each of duration 2 minutes, are chosen at random. Let p denote the probability that in each of the five time intervals at most 1 call arrives at the telephone exchange. Then e^{10} p (in integer) is equal to ______

29. Let X be a random variable with the probability density function

where c is a constant and [x] denotes the greatest integer less than or equal to x. If A = [1/2, 2], then P(X ∈ A) (rounded off to two decimal places) is equal to ______

30. Let X and Y be two random variables such that the moment generating function of X is M(t) and the moment generating function of Y is where t ∈ (−h, h), h > 0. If the mean and the variance of X are 1/2 and 1/4, respectively, then the variance of Y (in integer) is equal to _______

31. Let X_{i,} i = 1, 2, … n, be i.i.d. random variables with the probability density function

where Γ(∙)denotes the gamma function. Also, let If converges to N(0, σ^{2}) in distribution, then σ^{2} (rounded off to two decimal places) is equal to ________

32. Consider a Poisson process {X(t), t ≥ 0}. The probability mass function of X(t) is given by

If C(t_{1}, t_{2}) is the covariance function of the Poisson process, then the value of C(5, 3) (in integer) is equal to ______

33. A random sample of size 4 is taken from the distribution with the probability density function

If the observed sample values are 6, 5, 3, 6, then the method of moments estimate (in integer) of the parameter θ, based on these observations, is ______

34. A company sometimes stops payments of quarterly dividends. If the company pays the quarterly dividend, the probability that the next one will be paid is 0.7. If the company stops the quarterly dividend, the probability that the next quarterly dividend will not be paid is 0.5. Then the probability (rounded off to three decimal places) that the company will not pay quarterly dividend in the long run is ______

35. Let X_{1}, X_{2}, …, X_{8} be a random sample taken from a distribution with the probability density function

Let F_{8}(x) be the empirical distribution function of the sample. If α is the variance of F_{8}(2), then 128α (in integer) is equal to ______

**Q.36 – Q.65 Carry TWO marks Each**

36. Let M be a 3 × 3 real symmetric matrix with eigenvalues −1, 1, 2 and the corresponding unit eigenvectors u, v, w respectively. Let x and y be two vectors in ℝ^{3} such that

Mx = u +2(v + w) and M^{2}y = u – (v + 2w).

Considering the usual inner product in ℝ^{3}, the value of |x + y|^{2}, where |x + y| is the length of vector x + y, is

(A) 1.25

(B) 0.25

(C) 0.75

(D) 1

37. Consider the following infinite series:

Which of the above series is/are conditionally convergent?

(A) S_{1} only

(B) S_{2} only

(C) Both S_{1} and S_{2}

(D) Neither S_{1} nor S_{2}

38. Let (3, 6)^{T}, (4, 4)^{T}, (5, 7)^{T} and (4, 7)^{T} be four independent observations from a bivariate normal distribution with the mean vector μ and the covariance matrix ∑. Let be the maximum likelihood estimates of μ and ∑, respectively, based on these observations. Then is equal to

39. Let follow N_{3}(μ, ∑) with where a ∈ ℝ. Suppose that the partial correlation coefficient between X_{2} and X_{3}, keeping X_{1} fixed is 5/7. Then a is equal to

(A) 1

(B) 3/2

(C) 2

(D) 1/2

40. If the line y = αx, α ≥ √2, divides the area of the region

R: = {(x, y) ∈ ℝ^{2}| 0 ≤ x ≤ √y, 0 ≤ y ≤ 2}

into two equal parts, then the value of α is equal to

(A) 3/√2

(B) 2√2

(C) √2

(D) 5/2√2

41. Let (X, Y, Z) be a random vector with the joint probability density function

Then which one of the following points is on the regression surface of X on (Y, Z) ?

42. A random sample X of size one is taken from a distribution with the probability density function

If x/θ is used as a pivot for obtaining the confidence interval for θ, hen which one of the following is an 80% confidence interval (confidence limits rounded off to three decimal places) for θ based on the observed sample value x = 10?

(A) (10.541, 31.623)

(B) (10.987, 31.126)

(C) (11.345, 30.524)

(D) (11.267, 30.542)

43. Let X_{1}, X_{2}, …, X_{7} be a random sample from a normal population with mean 0 and variance θ > 0. Let

Consider the following statements:

(I) The statistics K and K_{1}^{2} + X_{2}^{2} + … + X_{7}^{2} are independent.

(II) 7K/2 has an F-distribution with 2 and 7 degrees of freedom.

(III) E(K^{2}) = 8/63.

Then which of the above statements is/are true?

(A) (I) and (II) only

(B) (I) and (III) only

(C) (II) and (III) only

(D) (I) only

44. Consider the following statements:

(I) Let a random variable X have the probability density function Then there exist i.i.d. random variables X_{1} and X_{2} such that X and X_{1} – X_{2} have the same distribution.

(II) Let a random variable Y have the probability density function

Then there exist i.i.d. random variables Y_{1} and Y_{2} such that Y and Y_{1} – Y_{2} have the same distribution.

Then which of the above statements is/are true?

(A) (I) only

(B) (II) only

(C) Both (I) and (II)

(D) Neither (I) nor (II)

45. Suppose X_{1}, X_{2}, …, X_{n}, … are independent exponential random variables with the mean 1/2. Let the notation o. denote ‘infinitely often’. Then which of the following is/are true?

46. Let {X_{n}}, n ≥ 1, be a sequence of random variables with the probability with the probability mass functions

Let X be a random variable with P(X = 0) = 1. Then which of the following statements is/are true?

(A) X_{n} converges to X in distribution

(B) X_{n} converges to X in probability

(C) E(X_{n}) → E(X)

(D) There exists a subsequence of {X_{n}} such that converges to X almost surely

47. Let M be any 3 × 3 symmetric matrix with eigenvalues 1, 2 and 3. Let N be any 3 × 3 matrix with real eigenvalues such that MN + N^{T}M = 3I, where I is the 3 × 3 identity matrix. Then which of the following cannot be eigenvalue(s) of the matrix N ?

(A) 1/4

(B) 3/4

(C) 1/2

(D) 7/4

48. Let M be a 3 × 2 real matrix having a singular value decomposition as M = USV^{T}, where the matrix U is a 3 × 3 orthogonal matrix, and V is a 2 × 2 orthogonal matrix. Then which of the following statements is/are true?

(A) The rank of the matrix M is 1

(B) The trace of the matrix M^{T}M is 4

(C) The largest singular value of the matrix (M^{T}M)^{−}^{1}M^{T} is 1

(D) The nullity of the matrix M is 1

49. Let X be a random variable such that

where ℤ denotes the set of all integers. If ϕ_{x}(t), t ∈ ℝ, denotes the characteristic function of X, then which of the following is/are true?

(A) ϕ_{x}(a) = 1

(B) ϕ_{x}(∙) is periodic with period a

(C) | ϕ_{x}(t)| < 1 for al t ≠ a

(D)

50. Which of the following real valued functions is/are uniformly continuous on [0, ∞)?

(A) sin^{2}x

(B) x sin x

(C) sin(sin x)

(D) sin(x sin x)

51. Two independent random samples, each of size 7, from two populations yield the following values:

If Mann-Whitney U test is performed at 5% level of significance to test the null hypothesis H_{0}: Distributions of the populations are same, against the alternative hypothesis H_{1}: Distributions of the populations are not same, then the value of the test statistic U (in integer) for the given data, is ______

52. Consider the multiple regression model

Y = β_{0} + β_{1}X_{1} + β_{2}X_{2} + β_{3}X_{3} + ϵ,

where ϵ is normally distributed with mean 0 and variance σ^{2} > 0, and β_{0}, β_{1}, β_{2}, β_{3} are unknown parameters. Suppose 52 observations of (Y, X_{1}, X_{2}, X_{3}) yield sum of squares due to regression as 18.6 and total sum of squares as 79.23. Then, for testing the null hypothesis H_{0} : β_{1} = β_{2} = β_{3} = 0 against the alternative hypothesis H_{1} : β_{i} ≠ 0 for some i = 1, 2, 3, the value of the test statistic (rounded off to three decimal places), based on one way analysis of variance, is _______

53. Suppose a random sample of size 3 is taken from a distribution with the probability density function

If p is the probability that the largest sample observation is at least twice the smallest sample observation, then the value of p (rounded off to three decimal places) is ______

54. Let a linear model Y = β_{0} + β_{1}X + ϵ be fitted to the following data, where ϵ is normally distributed with mean 0 and unknown variance σ^{2} > 0.

Let denote the ordinary least-square estimator of at X = 6, and the variance of Then the value of the real constant c (rounded off to one decimal place) is equal to ______

55. Let 0, 1, 1, 2, 0 be five observations of a random variable X which follows a Poisson distribution with the parameter θ > 0. Let the minimum variance unbiased estimate of P(X ≤ 1), based on this data, be α. Then 5^{4}α (in integer) is equal to ______

56. While calculating Spearman’s rank correlation coefficient, based on n observations {(x_{i}, y_{i}), i = 1, 2 …, n} from a paired data, it is found that x_{i} are distinct for all i ≥ 2, x_{1} = x_{2} and where d_{i} = rank(x_{i}) – rank(y_{i}). Then the minimum possible value of n^{3} – n (in integer) is ________

57. In a laboratory experiment, the behavior of cats are studied for a particular food preference between two foods A and B. For an experiment, 70% of the cats that had food A will prefer food A, and 50% of the cats that had food B will prefer food A. The experiment is repeated under identical conditions. If 40% of the cats had food A in the first experiment, then the percentage (rounded off to one decimal place) of cats those will prefer food A in the third experiment, is ______

58. A random sample of size 5 is taken from a distribution with the probability density function

where θ is an unknown parameter. If the observed values of the random sample are 3, 6, 4, 7, 5, then the maximum likelihood estimate of the 1/8^{th} quantile of the distribution (rounded off to one decimal place) is ____

59. Consider a gamma distribution with the probability density function

with β > 0. Then, for β = 2, the value of the Cramer-Rao lower bound (rounded off to one decimal place) for the variance of any unbiased estimator of β^{2}, based on a random sample of size 8 from this distribution, is ______

60. Let X_{1}, X_{2}, X_{3}, X_{4} be a random sample of size four from a Bernoulli distribution with the parameter θ, 0 < θ < 1. Consider the null hypothesis H_{0} : θ = 1/4 against the alternative hypothesis H_{1} : θ > 1/4. Suppose H_{0} is rejected if and only if X_{1} + X_{2} + X_{3} + X_{4} > 2. If α is the probability of Type I error for the test and γ(θ) is the power function of the test, then the value of 16α + 7γ(1/2) (in integer) is equal to _______

61. Given that Φ(1.645) = 0.95 and Φ(2.33) = 0.99, where Φ(⋅) denotes the cumulative distribution function of a standard normal random variable. For a random sample X_{1}, X_{2}, …, X_{n} from a normal population N(μ, 2^{2}), where μ is unknown, the null hypothesis H_{0} : μ = 10 to be tested against the alternative hypothesis H_{1} : μ = 12. Suppose that a test that rejects H_{0} if the sample mean is large, is used. Then the smallest value of n (in integer) such that Type I error is 0.05 and Type II error is at most 0.01, is ______

62. Let Y_{1} < Y_{2} < … < Y_{n} be the order statistics of a random sample of size n from a continuous distribution, which is symmetric about its mean μ. Then the smallest value of n (in integer) such that P(Y_{1} < μ < Y_{n}) ≥99, is ______

63. If P(x, y, z) is a point which is nearest to the origin and lies on the intersection of the surfaces z = xy + 5 and x + y + z = 1. Then the distance (in integer) between the origin and the point P is ______

64. Let X and Y be random variables such that X is uniformly distributed over (0, 4), and the conditional distribution of Y given X = x is uniformly distributed over (0, x^{2}/4). Then E(Y^{2}) (rounded off to three decimal places) is equal to _____

65. Let X = (X_{1}, X_{2}, X_{3})^{T} be a random vector with the distribution N_{3}(μ, Σ), where

Then E(X_{1}|(X_{2} = 4, X_{3} = 7)) (in integer) is equal to _____

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