GATE 2020 Statistics Previous Year Paper

GATE 2020 Statistics Previous Year Paper

GA – General Aptitude

Q1 – Q5 carry one mark each.

Q.No. 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,

(B)at on,

(C)on, at

(D)to, at

Q.No. 2 Despite a string of poor performances, the chances of K. L. Rahul’s selection in the team are

(A) slim

(B) bright

(D) uncertain

Q.No. 3 Select the word that fits the analogy:

Cover : Uncover :: Associate :

(A) Unassociate

(B) Inassociate

(D) Dissociate

Q.No. 4 Hit by floods, the kharif (summer sown) crops in various parts of the country 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.

Q.No. 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

(B) n² + n

(D) 2n² +n

Q6 – Q10 carry two marks each.

Q.No. 6 Repo rate is the rate 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.

Q.No. 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 neighbour of S.

Which of the following must be true?

(A) P is a neighbour of R.

(B) Q is a neighbour of R.

(D) R is the left neighbour of S.

Q.No. 8 The distance between Delhi and Agra is 233 km. A car P started travelling from Delhi to Agra and another car 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 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

(D) 116.5

Q. No. 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

(D) 16

Q.No. 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 Company P Company Q

(A) 15:17

(B) 16:17

(D) 17:16

Q1 – Q25 carry one mark each.

Q.No. 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 FALSE?

(A) The eigenvalues of M are 0 and 1

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

(D) (I₃ + M)^-1 = I₃ – 2M

Q.No.2 Let C denote the set of all complex numbers. Consider the vector space

V = {(a,b,c): a,b,c e C, a + 5 = 0, b + c = 0},

over the field of real numbers, where for any complex number z, 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,1,1)}

(D){(1, -i,1),(i,1,-1)}

Q.No. 3 Let S = {(x, y) E R R: x² – y² = 4} and f: S ⟶ R be defined by f(x,y) = 6 x + y2, where R denotes the set of all real numbers. Then

(A) f is bounded on S

(B) the maximum value off on S is 13

(D) the minimum value off on S is -13

Q.No. 4 Let f: R XR → R be defined by

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

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

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

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

Q.No. 5 The moment generating function of a random variable X is given by

Then P(X ≤ 2) equals

(A) 1/3

(B) 1/6

(D)5/6

Q.No. 6 Consider the following two-way fixed effects analysis of variance model

Where ε_ijk‘s are independently and identically distributed N(0,0%) random variables, .

Let SSE denote the sum of squares due to error. For any positive integer v and any ∝ € (0,1), let X_V²∝ denote the (1 – ∝)-th quantile of the central chi-square distribution with v degrees of freedom. Then a 95% confidence interval for σ² is given by SSE x?

Q.No. 7 Let X₁,…,X₂₀ be independent and identically distributed random variables with the common probability density function

Then the distribution of the random variable is

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

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

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

Q.No. 8 Let X₁, …, X₁₀ be a random sample from a Weibull distribution with the probability density function

where For any positive integer v and any a the quantile of the central chi-square distribution with v degrees of freedom. Then, a 90% confidence interval for θ is

Q.No. 9 Let X₁, …, X_n be a random sample of size n (≥ 2) from a uniform distribution on the interval [-θ, θ], where θ ε (θ,∞). A minimal sufficient statistic for θ is

Q.No. 10 Let X₁, …, X_n be a random sample of size n (≥ 2) from N(θ,2θ²) distribution, where θ ε (0,∞). Which of the following statements is TRUE?

Q.No. 11 Let 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

(D) 3

Q.No. 12 Consider the regression model

where β₀ and β₁ are unknown parameters and εi’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 β₁ is

Q.No. 13 Let In be a random sample of size n (≥2) from N_p (0, Σ) distribution, where is a positive definite matrix. Define

where for any column vector U, U^t denotes its transpose. Then the distribution of the statistic is

Q.No. 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

(D) 11

Q.No. 15 Let X₁, …, X_n be a random sample of size n (≥2) from an exponential distribution with the probability density function

where Consider the problem of testing against based on X₁, …, X_n. Which of the following statements is TRUE?

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

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

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

Q.No. 16 The characteristic function of a random variable X is given by

Q.No. 17 Let the random vector X = (X₁, X₂, X₃, X₄) follow N₄(μ, ∑) distribution, where

Then P(X₁ + X₂ + X₃ + X₄ > 0) = _______________ (correct up to one decimal place).

Q.No. 18 Let {X_n}_n_≥₀ be a homogeneous Markov chain with state space {0,1} and one

step transition probability matrix , then 27 × E(X₂) ______________= (correct up to two decimal places).

Q. No. 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).

Q.No. 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 variable Y + Z equals _____________ (correct up to one decimal place).

Q.No. 21 The total number of standard 4 X 4 Latin squares is___________

Q.No. 22 Let X be a 4 × 1 random vector with E(X) = 0 and variance-covariance matrix

Let Y 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).

Q.No. 23 Let X₁, …,X_n be a random sample of size n (≥2) from an exponential distribution with the probability density function

Q.No. 24 Let where x_i‘s are fixed covariates and εi’s are independent and identically distributed random variables with mean zero and finite variance. Suppose that α and β 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 and equals

Q.No. 25 Let {0,1,2,3} be an observed sample of size 4 from N(θ,5) distribution, where θ ε [2,∞). Then the maximum likelihood estimate of θ based on the observed sample is _________.

Q26 – Q55 carry two marks each.

Q.No. 26 Let f: $\mathbb{R}$ × $\mathbb{R}$ → $\mathbb{R}$ be defined by

where $\mathbb{R}$ denotes the set of all real numbers. Then

Q.No. 27 Consider the linear transformation T: $\mathbb{C}$ ³ → $\mathbb{C}$ ³ defined by

Where $\mathbb{C}$ is the set of all complex numbers and $\mathbb{C}$ ³ = $\mathbb{C}$ × $\mathbb{C}$ × $\mathbb{C}$ . 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

(D) T² = I₃, where I₃ is the 3 × 3 identity matrix

Q.No. 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 ∈ $\mathbb{R}$

(B) If a + c = 0 then M is diagonalizable for every b ∈ $\mathbb{R}$

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

Q.No. 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

(D) Trace(M) = 8

Q.No. 30 Let f: $\mathbb{R}$ × $\mathbb{R}$ → $\mathbb{R}$ be defined by

f(x.y)=|y-2||x-1|, (x,y) $\mathbb{R}$ × $\mathbb{R}$

where $\mathbb{R}$ 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)

(D) The directional derivative of f at (1,2) along u=12,12equals 1

Q.No. 31 Which of the following functions is uniformly continuous on the specified domain?

Q.No. 32 Let the random vector X = (X₁, X₂,X₃) 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

(D) Variance of X₁ + X₂ is π²

Q.No. 33 Suppose that P₁ and P₂ are two populations having bivariate normal distributions with mean vectors respectively, and the same variance-covariance matrix be 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

(D) both Z₁ and Z₂ are assigned to P₂

Q.No. 34 Let X₁, …, X_n be a random sample of size n (≥2) from an exponential distribution with the probability density function

where Which of the following statements is TRUE?

Q.No. 35. Let the joint distribution of (X, Y) be bivariate normal with mean vector and variance-covariance matrix where -1 < p < 1. Then E[max (X , Y)] equals

Q.No. 36 Let X₁, X₂, …, X₁₀ be independent and identically distributed N₃(0,I₃) 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 U, U^t 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

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

Q.No. 37 Let be independent and identically distributed random vectors, where is a positive definite matrix. Further, let 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 2, then which of the following statements is TRUE?

Q.No. 38 Let the joint distribution of the random variables 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

(D) X₁ – 2X₂ and 2X₁ + X₂ are independent

Q.No. 39 Consider the following one-way fixed effects analysis of variance model

Q.No. 40 Let X₁, …, X_n be a random sample of size from N(θ,1) distribution, where θ∈ (-∞,∞). Consider the problem of testing Ho: θ ∈ [1,2] against H:θ <1 or θ > 2, based on X₁, …,X_n. Which of the following statements is TRUE?

Q.No. 41 In a pure birth process with birth rates λ_n = 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) = _____________

Q.No. 42 Let {X_n]_n_≥₀ be a homogeneous Markov chain whose state space is {0,1,2} and TO 1 01 whose one-step transition probability matrix is Then nP(X2n=2|X0=2)= ________________(correct up to one decimal place).

Q.No. 43 Let (X,Y) be a random vector such that, for any y > 0, the conditional probability density function of X given Y = y is

If the marginal probability density function of Y is

then E(Y|X = 1) = ________ (correct up to one decimal place).

Q.No. 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 and Φ (1.5) = 0.9332 then the value of 2 p + 1 (round off to two decimal places) equals _______

Q.No. 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(X₀ = 1) = 1. Let p be the probability that state 0 will be visited before state 3. Then 6 × p = __________

Q.No. 46 Let (X,Y) be a random vector with joint probability mass function

where Then the variance of Y equals ____________

Q.No. 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).

Q.No. 48 Let X = (X₁,X₂,X₃) be a random vector following N₃ (0, ∑) distribution,

where . Then the partial correlation coefficient between X₂ and X₃, with fixed X₁, equals _______ (correct up to two decimal places).

Q.No. 49 Let X₁, X₂, X₃ and X₄ be a random sample from a population having probability density function and f(-x) = f(x), for all For testing H₀: θ = 0 against H₁: θ <0, let T⁺ denote the Wilcoxon Signed-rank statistic. Then under H₀,

Q.No. 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 _________

Q.No. 51 Let where $\mathbb{R}$ denotes the set of all real numbers. If then 84 × 1 = _____________________________

Q.No. 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) = _________________

Q.No. 53 Let {X_n}_n_≥₁ be a sequence of independent and identically distributed random variables with Further, let

where Φ (.) denotes the cumulative distribution function of the standard normal distribution, then c² = _______________(correct up to one decimal place).

Q.No. 54 Let the random vector X = (X₁, X₂, X₃) have the joint probability density function