GATE 2023 Question Paper ST Statistics

The Graduate Aptitude Test in Engineering

GATE 2023
Question Paper
GATE is an entrance examination conducted in India that
primarily tests the comprehensive understanding of
undergraduate subjects in engineering and sciences for
admission into technical postgraduate programs.

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General Aptitude (GA)
Q.1 – Q.5 Carry ONE mark Each

Q.1 “I have not yet decided what I will do this evening; I ______ visit a friend.”

(A) mite

(B) would

(C) might

(D) didn’t

Q.2 Eject : Insert : : Advance : _______

(By word meaning)

(A) Advent

(B) Progress

(C) Retreat

(D) Loan

Page 1 of 48

Q.3 In the given figure, PQRSTV is a regular hexagon with each side of length 5 cm. A
circle is drawn with its centre at V such that it passes through P. What is the area
(in cm2 ) of the shaded region? (The diagram is representative)

(A) 25π
3

(B) 20π
3

(C) 6π

(D) 7π

Page 2 of 48

Q.4 A duck named Donald Duck says “All ducks always lie.”

Based only on the information above, which one of the following statements can be
logically inferred with certainty?

(A) Donald Duck always lies.

(B) Donald Duck always tells the truth.

(C) Donald Duck’s statement is true.

(D) Donald Duck’s statement is false.

Page 3 of 48

Q.5 A line of symmetry is defined as a line that divides a figure into two parts in a way
such that each part is a mirror image of the other part about that line.

The figure below consists of 20 unit squares arranged as shown. In addition to the
given black squares, upto 5 more may be coloured black. Which one among the
following options depicts the minimum number of boxes that must be coloured
black to achieve two lines of symmetry? (The figure is representative)

a b

c d

e f g h

i

j k

(A) d

(B) c, d, i

(C) c, i

(D) c, d, i, f, g

Page 4 of 48

Q.6 – Q.10 Carry TWO marks Each

Q.6 Based only on the truth of the statement ‘Some humans are intelligent’, which one
of the following options can be logically inferred with certainty?

(A) No human is intelligent.

(B) All humans are intelligent.

(C) Some non-humans are intelligent.

(D) Some intelligent beings are humans.

Page 5 of 48

Q.7 Which one of the options can be inferred about the mean, median, and mode for the
given probability distribution (i.e. probability mass function), 𝑃(𝑥), of a variable x?

(A) mean = median ≠ mode

(B) mean = median = mode

(C) mean ≠ median = mode

(D) mean ≠ mode = median

Page 6 of 48

Q.8 The James Webb telescope, recently launched in space, is giving humankind
unprecedented access to the depths of time by imaging very old stars formed almost
13 billion years ago. Astrophysicists and cosmologists believe that this odyssey in
space may even shed light on the existence of dark matter. Dark matter is supposed
to interact only via the gravitational interaction and not through the
electromagnetic-, the weak- or the strong-interaction. This may justify the epithet
“dark” in dark matter.

Based on the above paragraph, which one of the following statements is FALSE?

(A) No other telescope has captured images of stars older than those captured by the
James Webb telescope.

(B) People other than astrophysicists and cosmologists may also believe in the existence
of dark matter.

(C) The James Webb telescope could be of use in the research on dark matter.

(D) If dark matter was known to interact via the strong-interaction, then the epithet
“dark” would be justified.

Page 7 of 48

Q.9 Let 𝑎 = 30! , 𝑏 = 50! , and 𝑐 = 100! . Consider the following numbers:

log 𝑎 𝑐, log 𝑐 𝑎, log 𝑏 𝑎, log 𝑎 𝑏

Which one of the following inequalities is CORRECT?

(A) log 𝑐 𝑎 < log 𝑏 𝑎 < log 𝑎 𝑏 < log 𝑎 𝑐

(B) log 𝑐 𝑎 < log 𝑎 𝑏 < log 𝑏 𝑎 < log 𝑏 𝑐

(C) log 𝑐 𝑎 < log 𝑏 𝑎 < log 𝑎 𝑐 < log 𝑎 𝑏

(D) log 𝑏 𝑎 < log 𝑐 𝑎 < log 𝑎 𝑏 < log 𝑎 𝑐

Page 8 of 48

Q.10 A square of side length 4 cm is given. The boundary of the shaded region is defined
by one semi-circle on the top and two circular arcs at the bottom, each of radius
2 cm, as shown.

The area of the shaded region is _______ cm2 .

(A) 8

(B) 4

(C) 12

(D) 10

Page 9 of 48

Q.11 – Q.35 Carry ONE mark Each

Q.11 The area of the region bounded by the parabola 𝑥 = −𝑦 2 and the line
𝑦 = 𝑥 + 2 equals

(A) 3
2

(B) 7
2

(C) 9
2

(D) 9

Q.12 Let 𝐴 be a 3 × 3 real matrix having eigenvalues 1, 0, and −1.
If 𝐵 = 𝐴2 + 2𝐴 + 𝐼3 , where 𝐼3 is the 3 × 3 identity matrix, then which one
of the following statements is true?

(A) 𝐵3 − 5𝐵 2 + 4𝐵 = 0

(B) 𝐵3 − 5𝐵 2 − 4𝐵 = 0

(C) 𝐵3 + 5𝐵 2 − 4𝐵 = 0

(D) 𝐵3 + 5𝐵 2 + 4𝐵 = 0

Page 10 of 48

Q.13 Consider the following statements.

(I) Let 𝐴 and 𝐵 be two 𝑛 × 𝑛 real matrices. If 𝐵 is invertible, then
𝑟𝑎𝑛𝑘 (𝐵𝐴) = 𝑟𝑎𝑛𝑘(𝐴).
(II) Let 𝐴 be an 𝑛 × 𝑛 real matrix. If 𝐴2 𝒙 = 𝒃 has a solution for every
𝒃 ∈ ℝ𝑛 , then 𝐴𝒙 = 𝒃 also has a solution for every 𝒃 ∈ ℝ𝑛 .

Which of the above statements is/are true?

(A) Only (I)

(B) Only (II)

(C) Both (I) and (II)

(D) Neither (I) nor (II)

Page 11 of 48

Q.14 Consider the probability space (Ω, 𝒢, 𝑃 ), where Ω = [0, 2] and
𝒢 = {𝜙, Ω, [0,1], (1,2]}. Let 𝑋 and 𝑌 be two functions on Ω defined as

1 𝑖𝑓 𝜔 ∈ [0, 1]
𝑋 (𝜔 ) = {
2 𝑖𝑓 𝜔 ∈ (1, 2]

and

2 𝑖𝑓 𝜔 ∈ [0, 1.5]
𝑌 (𝜔 ) = {
3 𝑖𝑓 𝜔 ∈ (1.5, 2].
Then which one of the following statements is true?

(A) 𝑋 is a random variable with respect to 𝒢, but 𝑌 is not a random variable with
respect to 𝒢

(B) 𝑌 is a random variable with respect to 𝒢, but 𝑋 is not a random variable with
respect to 𝒢

(C) Neither 𝑋 nor 𝑌 is a random variable with respect to 𝒢

(D) Both 𝑋 and 𝑌 are random variables with respect to 𝒢

Page 12 of 48

Q.15 Let Φ(⋅) denote the cumulative distribution function of a standard normal
random variable. If the random variable 𝑋 has the cumulative distribution
function

Φ(𝑥) 𝑖𝑓 𝑥 < −1
𝐹(𝑥) = {
Φ(𝑥 + 1) 𝑖𝑓 𝑥 ≥ −1,
then which one of the following statements is true?

(A) 1
𝑃(𝑋 ≤ −1) =
2

(B) 1
𝑃(𝑋 = −1) =
2

(C) 1
𝑃(𝑋 < −1) =
2

(D) 1
𝑃(𝑋 ≤ 0) =
2

Page 13 of 48

Q.16 Let 𝑋 be a random variable with probability density function
𝛼
𝛼𝜆 𝑥 𝛼−1 𝑒 −𝜆𝑥 𝑖𝑓 𝑥 > 0
𝑓(𝑥) = {
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒,
where 𝛼 > 0 and 𝜆 > 0. If the median of 𝑋 is 1 and the third quantile is 2,
then (𝛼, 𝜆) equals

(A) (1, log𝑒 2)

(B) (1, 1)

(C) (2, log𝑒 2)

(D) (1, log 𝑒 3)

Page 14 of 48

Q.17 Let 𝑋 be a random variable having Poisson distribution with mean 𝜆 > 0. Then
1
𝐸( | 𝑋 > 0) equals
𝑋+1

(A) 1 − 𝑒 −𝜆 − 𝜆𝑒 −𝜆
𝜆(1 − 𝑒 − 𝜆 )

(B) 1 − 𝑒 −𝜆
𝜆

(C) 1 − 𝑒 −𝜆 − 𝜆𝑒 −𝜆
𝜆

(D) 1 − 𝑒 −𝜆
𝜆+1

Page 15 of 48

Q.18 Suppose that 𝑋 has the probability density function

𝜆𝛼 𝛼−1 −𝜆𝑥
𝑥 𝑒 𝑖𝑓 𝑥 > 0
𝑓 (𝑥) = {Γ(𝛼)
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒,

where 𝛼 > 0 and 𝜆 > 0. Which one of the following statements is NOT true?

(A) 𝐸 (𝑋 ) exists for all 𝛼 > 0 and 𝜆 > 0

(B) Variance of 𝑋 exists for all 𝛼 > 0 and 𝜆 > 0

(C) 1
𝐸 ( ) exists for all 𝛼 > 0 and 𝜆 > 0
𝑋

(D) 𝐸 (log 𝑒 (1 + 𝑋 )) exists for all 𝛼 > 0 and 𝜆 > 0

Page 16 of 48

Q.19 Let (𝑋, 𝑌) have joint probability density function

8𝑥𝑦 𝑖𝑓 0 < 𝑥 < 𝑦 < 1
𝑓 (𝑥, 𝑦) = {
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.
1
If 𝐸 (𝑋 | 𝑌 = 𝑦0 ) = , then 𝑦0 equals
2

(A) 3
4

(B) 1
2

(C) 1
3

(D) 2
3

Page 17 of 48

Q.20 Suppose that there are 5 boxes, each containing 3 blue pens, 1 red pen and 2
black pens. One pen is drawn at random from each of these 5 boxes. If the random
variable 𝑋1 denotes the total number of blue pens drawn and the random variable
𝑋2 denotes the total number of red pens drawn, then 𝑃(𝑋1 = 2, 𝑋2 = 1) equals

(A) 5
36

(B) 5
18

(C) 5
12

(D) 5
9

Q.21 Let {𝑋𝑛 }𝑛≥1 and {𝑌𝑛 }𝑛≥1 be two sequences of random variables and 𝑋 and 𝑌
be two random variables, all of them defined on the same probability space.
Which one of the following statements is true?

(A) If {𝑋𝑛 }𝑛≥1 converges in distribution to a real constant 𝑐 , then {𝑋𝑛 }𝑛≥1
converges in probability to 𝑐

(B) If {𝑋𝑛 }𝑛≥1 converges in probability to 𝑋, then {𝑋𝑛 }𝑛≥1 converges in 3rd mean
to 𝑋

If {𝑋𝑛 }𝑛≥1 converges in distribution to 𝑋 and {𝑌𝑛 }𝑛≥1 converges in

(C)
distribution to 𝑌, then {𝑋𝑛 + 𝑌𝑛 }𝑛≥1 converges in distribution to 𝑋 + 𝑌

(D) If {𝐸(𝑋𝑛 )}𝑛≥1 converges to 𝐸(𝑋), then {𝑋𝑛 }𝑛≥1 converges in 1st mean to 𝑋

Page 18 of 48

Q.22 Let 𝑋 be a random variable with probability density function

1 −𝑥
𝑓(𝑥; 𝜆) = { 𝜆 𝑒 𝑖𝑓 𝑥 > 0
𝜆

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒,
where 𝜆 > 0 is an unknown parameter. Let 𝑌1 , 𝑌2 , … , 𝑌𝑛 be a random sample of
size 𝑛 from a population having the same distribution as 𝑋 2 .
1
̅ = ∑𝑛𝑖=1 𝑌𝑖 , then which one of the following statements is true?
If 𝑌
𝑛

(A) 𝑌̅
√ is a method of moments estimator of 𝜆
2

(B) √𝑌̅ is a method of moments estimator of 𝜆

(C) 1
√𝑌̅ is a method of moments estimator of 𝜆
2

(D) 2√𝑌̅ is a method of moments estimator of 𝜆

Page 19 of 48

Q.23 Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 be a random sample of size 𝑛 (≥ 2) from a population
having probability density function

2 (log𝑒 𝑥)2
−
( )
𝑓 (𝑥; 𝜃) = { 𝜃𝑥 − log 𝑒 𝑥 𝑒 𝑖𝑓 0 < 𝑥 < 1
𝜃

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒,
where 𝜃 > 0 is an unknown parameter. Then which one of the following
statements is true?

(A) 1
∑𝑛𝑖=1(log 𝑒 𝑋𝑖 )2 is the maximum likelihood estimator of 𝜃
𝑛

(B) 1
∑𝑛𝑖=1(log 𝑒 𝑋𝑖 )2 is the maximum likelihood estimator of 𝜃
𝑛−1

(C) 1
∑𝑛𝑖=1 log 𝑒 𝑋𝑖 is the maximum likelihood estimator of 𝜃
𝑛

(D) 1
∑𝑛𝑖=1 log 𝑒 𝑋𝑖 is the maximum likelihood estimator of 𝜃
𝑛−1

Page 20 of 48

Q.24 Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 be a random sample of size 𝑛 from a population having
1 1
uniform distribution over the interval ( , 𝜃), where 𝜃 > is an unknown
3 3
parameter. If 𝑌 = max{𝑋1 , 𝑋2 , … , 𝑋𝑛 }, then which one of the following
statements is true?

(A) 𝑛+1 1 1
( ) (𝑌 − ) + is an unbiased estimator of 𝜃
𝑛 3 3

(B) 𝑛 1 1
( ) (𝑌 − ) + is an unbiased estimator of 𝜃
𝑛+1 3 3

(C) 𝑛+1 1 1
( ) (𝑌 + ) − is an unbiased estimator of 𝜃
𝑛 3 3

(D) 𝑌 is an unbiased estimator of 𝜃

Page 21 of 48

Q.25 Suppose that 𝑿1 , 𝑿2 , … , 𝑿𝑛 , 𝒀1 , 𝒀2 , … , 𝒀𝑛 are independent and identically
distributed random vectors each having 𝑁𝑝 (𝝁, Σ) distribution, where Σ is non-
1 1
singular, 𝑝 > 1 and 𝑛 > 1. If 𝑿 ̅ = ∑𝑛𝑖=1 𝑿𝑖 and 𝒀
̅ = ∑𝑛𝑖=1 𝒀𝑖 , then which
𝑛 𝑛
one of the following statements is true?

(A) ̅ − 𝝁)𝑇 Σ−1 (𝑿
There exists 𝑐 > 0 such that 𝑐 (𝑿 ̅ − 𝝁) has 𝜒 2 -distribution
with 𝑝 degrees of freedom

(B) ̅−𝒀
There exists 𝑐 > 0 such that 𝑐 (𝑿 ̅ )𝑇 Σ−1 (𝑿
̅−𝒀
̅ ) has 𝜒 2 -distribution
with (𝑝 − 1) degrees of freedom

(C) There exists 𝑐 > 0 such that 𝑐 ∑𝑛𝑖=1(𝑿𝑖 − 𝑿 ̅ )𝑇 Σ −1 (𝑿𝑖 − 𝑿
̅ ) has
𝜒 2 -distribution with 𝑝 degrees of freedom

(D) ̅+𝒀
There exists 𝑐 > 0 such that 𝑐 ∑𝑛𝑖=1(𝑿𝑖 − 𝒀𝑖 − 𝑿 ̅ )𝑇 Σ −1 (𝑿𝑖 − 𝒀𝑖 − 𝑿
̅+𝒀
̅)
2
has 𝜒 -distribution with 𝑝 degrees of freedom

Page 22 of 48

Q.26 Consider the following regression model

𝑦𝑘 = 𝛼0 + 𝛼1 log 𝑒 𝑘 + 𝜖𝑘 , 𝑘 = 1, 2, … , 𝑛,

where 𝜖𝑘 ’s are independent and identically distributed random variables each
1
having probability density function 𝑓 (𝑥) = 𝑒 −|𝑥| , 𝑥 ∈ ℝ. Then which one of
2
the following statements is true?

(A) The maximum likelihood estimator of 𝛼0 does not exist

(B) The maximum likelihood estimator of 𝛼1 does not exist

(C) The least squares estimator of 𝛼0 exists and is unique

(D) The least squares estimator of 𝛼1 exists, but it is not unique

Page 23 of 48

Q.27 Suppose that 𝑋1 , 𝑋2 , … , 𝑋𝑛 are independent and identically distributed random
variables each having probability density function 𝑓(⋅) and median 𝜃 . We want
to test

𝐻0 : 𝜃 = 𝜃0 against 𝐻1 : 𝜃 > 𝜃0 .

Consider a test that rejects 𝐻0 if 𝑆 > 𝑐 for some 𝑐 depending on the size of
the test, where 𝑆 is the cardinality of the set {𝑖: 𝑋𝑖 > 𝜃0 , 1 ≤ 𝑖 ≤ 𝑛}. Then
which one of the following statements is true?

(A) Under 𝐻0 , the distribution of 𝑆 depends on 𝑓(⋅)

(B) Under 𝐻1 , the distribution of 𝑆 does not depend on 𝑓(⋅)

(C) The power function depends on 𝜃

(D) The power function does not depend on 𝜃

Page 24 of 48

Q.28 Suppose that 𝑥 is an observed sample of size 1 from a population with
probability density function 𝑓(⋅). Based on 𝑥 , consider testing

1 𝑦2 1 −|𝑦|
𝐻0 : 𝑓 (𝑦) = 𝑒− 2 ; 𝑦 ∈ ℝ against 𝐻1 : 𝑓(𝑦) = 𝑒 ; 𝑦 ∈ ℝ.
√2𝜋 2

Then which one of the following statements is true?

(A) The most powerful test rejects 𝐻0 if |𝑥| > 𝑐 for some 𝑐 > 0

(B) The most powerful test rejects 𝐻0 if |𝑥| < 𝑐 for some 𝑐 > 0

(C) The most powerful test rejects 𝐻0 if ||𝑥| − 1| > 𝑐 for some 𝑐 > 0

(D) The most powerful test rejects 𝐻0 if ||𝑥| − 1| < 𝑐 for some 𝑐 > 0

Q.29 Let 𝑓: ℝ2 → ℝ be defined by 𝑓 (𝑥, 𝑦) = 𝑥𝑦. Then the maximum value (rounded
off to two decimal places) of 𝑓 on the ellipse 𝑥 2 + 2𝑦 2 = 1
equals _______________

Q.30 Let 𝐴 be a 2 × 2 real matrix such that 𝐴𝐵 = 𝐵𝐴 for all 2 × 2 real matrices
𝐵. If trace of 𝐴 equals 5, then determinant of 𝐴 (rounded off to two decimal
places) equals _______________

Q.31 Two defective bulbs are present in a set of five bulbs. To remove the two
defective bulbs, the bulbs are chosen randomly one by one and tested. If 𝑋

denotes the minimum number of bulbs that must be tested to find out the two
defective bulbs, then 𝑃 (𝑋 = 3) (rounded off to two decimal places)
equals _______________

Page 25 of 48

Q.32 Let {𝑋𝑛 }𝑛≥1 be a sequence of independent and identically distributed random
1
variables each having mean 4 and variance 9. If 𝑌𝑛 = ∑𝑛𝑖=1 𝑋𝑖 for 𝑛 ≥ 1,
𝑛
𝑌𝑛 −4 2
then lim 𝐸 [( ) ] (in integer) equals _______________
𝑛→∞ √𝑛

Q.33 Let {𝑊𝑡 }𝑡≥0 be a standard Brownian motion. Then 𝐸 (𝑊42 | 𝑊2 = 2)
(in integer) equals _______________

Q.34 Let {𝑋𝑛 }𝑛≥1 be a Markov chain with state space {1, 2, 3} and transition
probability matrix

1 1 1
2 4 4
1 1 1
.
3 3 3
1 1
[ 0
2 2]
Then 𝑃 (𝑋2 = 1 | 𝑋1 = 1, 𝑋3 = 2) (rounded off to two decimal places)
equals _______________

Q.35 0
Suppose that (𝑋1 , 𝑋2 , 𝑋3 ) has 𝑁3 (𝝁, Σ) distribution with 𝝁 = [0] and
0
2 2 1
Σ = [2 5 1].
1 1 1
Given that Φ(−0.5) = 0.3085, where Φ(⋅) denotes the cumulative
distribution function of a standard normal random variable,
7

𝑃 ((𝑋1 − 2𝑋2 + 2𝑋3 )2 < ) (rounded off to two decimal places)
2
equals _______________

Page 26 of 48

Q.36 – Q.65 Carry TWO marks Each

Q.36 Let 𝐴 be an 𝑛 × 𝑛 real matrix. Consider the following statements.

(I) If 𝐴 is symmetric, then there exists 𝑐 ≥ 0 such that 𝐴 + 𝑐𝐼𝑛 is
symmetric and positive definite, where 𝐼𝑛 is the 𝑛 × 𝑛 identity
matrix.
(II) If 𝐴 is symmetric and positive definite, then there exists a symmetric
and positive definite matrix 𝐵 such that 𝐴 = 𝐵 2 .

Which of the above statements is/are true?

(A) Only (I)

(B) Only (II)

(C) Both (I) and (II)

(D) Neither (I) nor (II)

Page 27 of 48

Q.37 Let 𝑋 be a random variable with probability density function

1
𝑓 (𝑥 ) = { 𝑥 2 𝑖𝑓 𝑥 ≥ 1
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.
If 𝑌 = log 𝑒 𝑋 , then 𝑃(𝑌 < 1 | 𝑌 < 2) equals

(A) 𝑒
1+𝑒

(B) 𝑒−1
𝑒+1

(C) 1
1+𝑒

(D) 1
𝑒−1

Page 28 of 48

Q.38 Let {𝑁(𝑡)}𝑡≥0 be a Poisson process with rate 1. Consider the following
statements.

3 3 2 2
(I) 𝑃(𝑁(3) = 3 | 𝑁(5) = 5) = (53) ( ) ( ) .
5 5
(II) If 𝑆5 denotes the time of occurrence of the 5th event for the above
Poisson process, then 𝐸 (𝑆5 | 𝑁(5) = 3) = 7.

Which of the above statements is/are true?

(A) Only (I)

(B) Only (II)

(C) Both (I) and (II)

(D) Neither (I) nor (II)

Page 29 of 48

Q.39 Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 be a random sample of size 𝑛 from a population having
probability density function

𝑒 −(𝑥−𝜇) 𝑖𝑓 𝜇 ≤ 𝑥 < ∞
𝑓 (𝑥; 𝜇) = {
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒,

where 𝜇 ∈ ℝ is an unknown parameter. If 𝑀 ̂ is the maximum likelihood
estimator of the median of 𝑋1 , then which one of the following statements is true?

(A) ̂ ≤ 2) = 1 − 𝑒 −𝑛(1−log𝑒 2) if 𝜇 = 1
𝑃(𝑀

(B) ̂ ≤ 1) = 1 − 𝑒 −𝑛 log𝑒 2 if 𝜇 = 1
𝑃(𝑀

(C) ̂ ≤ 3) = 1 − 𝑒 −𝑛(1−log𝑒 2) if 𝜇 = 1
𝑃(𝑀

(D) ̂ ≤ 4) = 1 − 𝑒 −𝑛(2 log𝑒 2−1) if 𝜇 = 1
𝑃(𝑀

Page 30 of 48

Q.40 Let 𝑋1 , 𝑋2 , … , 𝑋10 be a random sample of size 10 from a population having
𝑁(0, 𝜃 2 ) distribution, where 𝜃 > 0 is an unknown parameter.
1
Let 𝑇 = ∑10 2
𝑖=1 𝑋𝑖 . If the mean square error of 𝑐𝑇 (𝑐 > 0), as an estimator of
10
𝜃 2 , is minimized at 𝑐 = 𝑐0 , then the value of 𝑐0 equals

(A) 5
6

(B) 2
3

(C) 3
5

(D) 1
2

Page 31 of 48

Q.41 Suppose that 𝑿1 , 𝑿2 , … , 𝑿10 are independent and identically distributed random
vectors each having 𝑁2 (𝝁, Σ) distribution, where Σ is non-singular. If

1
𝑈= ,
̅ − 𝝁)𝑇 Σ−1 (𝑿
1 + (𝑿 ̅ − 𝝁)
1 1
̅=
where 𝑿 ∑10
𝑖=1 𝑿𝑖 , then the value of log 𝑒 𝑃 (𝑈 ≤ ) equals
10 2

(A) −5

(B) −10

(C) −2

(D) −1

Page 32 of 48

Q.42 Suppose that (𝑋, 𝑌) has joint probability mass function

𝑃(𝑋 = 0, 𝑌 = 0) = 𝑃(𝑋 = 1, 𝑌 = 1) = 𝜃,
1
𝑃(𝑋 = 1, 𝑌 = 0) = 𝑃(𝑋 = 0, 𝑌 = 1) = − 𝜃,
2
1 1
where 0 ≤ 𝜃 ≤ is an unknown parameter. Consider testing 𝐻0 : 𝜃 =
2 4
1
against 𝐻1 : 𝜃 = , based on a random sample {(𝑋1 , 𝑌1 ), (𝑋2 , 𝑌2 ), … , (𝑋𝑛 , 𝑌𝑛 )}
3
from the above probability mass function. Let 𝑀 be the cardinality of the set
{𝑖: 𝑋𝑖 = 𝑌𝑖 , 1 ≤ 𝑖 ≤ 𝑛}. If 𝑚 is the observed value of 𝑀, then which one of the
following statements is true?

(A) The likelihood ratio test rejects 𝐻0 if 𝑚 > 𝑐 for some 𝑐

(B) The likelihood ratio test rejects 𝐻0 if 𝑚 < 𝑐 for some 𝑐

(C) The likelihood ratio test rejects 𝐻0 if 𝑐1 < 𝑚 < 𝑐2 for some 𝑐1 and 𝑐2

(D) The likelihood ratio test rejects 𝐻0 if 𝑚 < 𝑐1 or 𝑚 > 𝑐2 for some 𝑐1 and 𝑐2

Page 33 of 48

Q.43 Let 𝑔(𝑥) = 𝑓 (𝑥) + 𝑓(2 − 𝑥) for all 𝑥 ∈ [0, 2] , where 𝑓: [0, 2] → ℝ is
continuous on [0, 2] and twice differentiable on (0, 2). If 𝑔′ denotes the
derivative of 𝑔 and 𝑓 ′′ denotes the second derivative of 𝑓 , then which one of
the following statements is NOT true?

(A) There exists 𝑐 ∈ (0, 2) such that 𝑔′ (𝑐 ) = 0

(B) If 𝑓 ′′ > 0 on (0, 2), then 𝑔 is strictly decreasing on (0, 1)

(C) If 𝑓 ′′ < 0 on (0, 2), then 𝑔 is strictly increasing on (1, 2)

(D) If 𝑓 ′′ = 0 on (0, 2), then 𝑔 is a constant function

Q.44 For any subset 𝒰 of ℝ𝑛 , let 𝐿(𝒰) denote the span of 𝒰. For any two subsets
𝒯 and 𝒮 of ℝ𝑛 , which one of the following statements is NOT true?

(A) If 𝒯 is a proper subset of 𝒮 , then 𝐿(𝒯 ) is a proper subset of 𝐿(𝒮)

(B) 𝐿(𝐿(𝒮 )) = 𝐿(𝒮)

(C) 𝐿(𝒯 ∪ 𝒮 ) = {𝑢 + 𝑣: 𝑢 ∈ 𝐿(𝒯 ), 𝑣 ∈ 𝐿(𝒮)}

(D) If 𝛼, 𝛽 and 𝛾 are three vectors in ℝ𝑛 such that 𝛼 + 2𝛽 + 3𝛾 = 0, then
𝐿({𝛼, 𝛽}) = 𝐿({𝛽, 𝛾})

Page 34 of 48

Q.45 Let 𝑓 be a continuous function from [0, 1] to the set of all real numbers. Then
which one of the following statements is NOT true?

(A) ∞
𝑓(𝑥𝑛 )
For any sequence {𝑥𝑛 }𝑛≥1 in [0, 1], ∑ 2
is absolutely convergent
𝑛=1 𝑛

(B) 1
If |𝑓 (𝑥)| = 1 for all 𝑥 ∈ [0, 1], then | ∫0 𝑓(𝑥)𝑑𝑥 | = 1

(C) If {𝑥𝑛 }𝑛≥1 is a sequence in [0, 1] such that {𝑓 (𝑥𝑛 )}𝑛≥1 is convergent, then
{𝑥𝑛 }𝑛≥1 is convergent

(D) If 𝑓 is also monotonically increasing, then the image of 𝑓 is given by
[𝑓 (0), 𝑓(1)]

Page 35 of 48

Q.46 Let 𝑋 be a random variable with cumulative distribution function

0 𝑖𝑓 𝑥 < −1
1
(𝑥 + 1) 𝑖𝑓 − 1 ≤ 𝑥 < 0
𝐹 (𝑥 ) = 4
1
(𝑥 + 3) 𝑖𝑓 0 ≤ 𝑥 < 1
4
{ 1 𝑖𝑓 𝑥 ≥ 1.

Which one of the following statements is true?

(A) 1 1 1 5
lim 𝑃 (− + < 𝑋 < − ) =
𝑛→∞ 2 𝑛 𝑛 8

(B) 1 1 1 5
lim 𝑃 (− − < 𝑋 < ) =
𝑛→∞ 2 𝑛 𝑛 8

(C) 1 1
lim 𝑃 (𝑋 = ) =
𝑛→∞ 𝑛 2

(D) 1
𝑃(𝑋 = 0) =
3

Page 36 of 48

Q.47 Let (𝑋, 𝑌) have joint probability mass function
𝑐
𝑖𝑓 𝑥 = 0, 1, 2, … ; 𝑦 = 0, 1, 2, … ; 𝑥 ≠ 𝑦
𝑝(𝑥, 𝑦) = { 2𝑥+𝑦+2
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.
Then which one of the following statements is true?

(A) 1
𝑐=
2

(B) 1
𝑐=
4

(C) 𝑐>1

(D) 𝑋 and 𝑌 are independent

Page 37 of 48

Q.48 Let 𝑿1 , 𝑿2 , … , 𝑿10 be a random sample of size 10 from a 𝑁3 (𝝁, Σ)
distribution, where 𝝁 and non-singular Σ are unknown parameters. If
5 10
1 1
̅ 1 = ∑ 𝑿𝑖 ,
𝑿 ̅ 2 = ∑ 𝑿𝑖 ,
𝑿
5 5
𝑖=1 𝑖=6
5 10
1 1
𝑆1 = ̅ 1 )(𝑿𝑖 − 𝑿
∑ (𝑿 𝑖 − 𝑿 ̅ 1 )𝑇 , 𝑆2 = ∑(𝑿𝑖 − 𝑿
̅ 2 )(𝑿𝑖 − 𝑿
̅ 2 )𝑇 ,
4 4
𝑖=1 𝑖=6

then which one of the following statements is NOT true?

(A) 5
̅ 1 − 𝝁)𝑇 𝑆1−1 (𝑿
(𝑿 ̅ 1 − 𝝁) follows a 𝐹-distribution with 3 and 2 degrees of
6
freedom

(B) 6
follows a 𝐹-distribution with 2 and 3 degrees of
̅ 1 −𝝁)𝑇 𝑆−1
5 (𝑿 ̅ 1 −𝝁)
1 (𝑿
freedom

(C) 4(𝑆1 + 𝑆2 ) follows a Wishart distribution of order 3 with 8 degrees of
freedom

(D) 5(𝑆1 + 𝑆2 ) follows a Wishart distribution of order 3 with 10 degrees of
freedom

Page 38 of 48

Q.49 Which of the following sets is/are countable?

(A) The set of all functions from {1, 2, 3, … , 10} to the set of all rational numbers

(B) The set of all functions from the set of all natural numbers to {0, 1}

(C) The set of all integer valued sequences with only finitely many non-zero terms

(D) The set of all integer valued sequences converging to 1

Q.50 For a given real number 𝑎, let 𝑎+ = 𝑚𝑎𝑥{𝑎, 0} and 𝑎− = 𝑚𝑎𝑥 {−𝑎, 0} .
If {𝑥𝑛 }𝑛≥1 is a sequence of real numbers, then which of the following statements
is/are true?

(A) If {𝑥𝑛 }𝑛≥1 converges, then both {𝑥𝑛+ }𝑛≥1 and {𝑥𝑛− }𝑛≥1 converge

(B) If {𝑥𝑛 }𝑛≥1 converges to 0, then both {𝑥𝑛+ }𝑛≥1 and {𝑥𝑛− }𝑛≥1 converge to 0

(C) If both {𝑥𝑛+ }𝑛≥1 and {𝑥𝑛− }𝑛≥1 converge, then {𝑥𝑛 }𝑛≥1 converges

(D) If {𝑥𝑛2 }𝑛≥1 converges, then both {𝑥𝑛+ }𝑛≥1 and {𝑥𝑛− }𝑛≥1 converge

Page 39 of 48

Q.51 0 4 1 0
Let 𝐴 be a 3 × 3 real matrix such that 𝐴 [1] = [0] , 𝐴 [0] = [4] and
1 0 1 0
1 0
𝐴 [1] = [0]. Then which of the following statements is/are true?
0 4

(A) 1 2
𝐴 [ 0] = [ 2 ]
0 −2

(B) 0 2
𝐴 [1] = [−2]
0 2

(C) 1 2
𝐴 [1] = [0]
1 2

(D) 1 8
𝐴 [2] = [4]
3 0

Page 40 of 48

Q.52 Let 𝑋 be a positive valued continuous random variable with finite mean.
If 𝑌 = [𝑋 ], the largest integer less than or equal to 𝑋, then which of the
following statements is/are true?

(A) 𝑃(𝑌 ≤ 𝑢 ) ≤ 𝑃(𝑋 ≤ 𝑢) for all 𝑢 ≥ 0

(B) 𝑃(𝑌 ≥ 𝑢 ) ≤ 𝑃(𝑋 ≥ 𝑢) for all 𝑢 ≥ 0

(C) 𝐸 (𝑋 ) < 𝐸(𝑌)

(D) 𝐸 (𝑋 ) > 𝐸(𝑌)

Q.53 Let 𝑋 be a random variable with probability density function

𝑒 −𝑥 𝑖𝑓 𝑥 ≥ 0
𝑓 (𝑥 ) = {
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.
For 𝑎 < 𝑏, if 𝑈(𝑎, 𝑏) denotes the uniform distribution over the interval (𝑎, 𝑏),
then which of the following statements is/are true?

(A) 𝑒 −𝑋 follows 𝑈(−1, 0) distribution

(B) 1 − 𝑒 −𝑋 follows 𝑈(0, 2) distribution

(C) 2𝑒 −𝑋 − 1 follows 𝑈(−1, 1) distribution

(D) The probability mass function of 𝑌 = [𝑋 ] is

𝑃(𝑌 = 𝑘 ) = (1 − 𝑒 −1 )𝑒 −𝑘 𝑓𝑜𝑟 𝑘 = 0, 1, 2, …,

where [𝑥] denotes the largest integer not exceeding 𝑥

Page 41 of 48

Q.54 Suppose that 𝑋 is a discrete random variable with the following probability mass
function

1
𝑃 (𝑋 = 0) = (1 + 𝑒 −1 )
2
𝑒 −1
𝑃 (𝑋 = 𝑘 ) = 𝑓𝑜𝑟 𝑘 = 1, 2, 3, … .
2 𝑘!
Which of the following statements is/are true?

(A) 𝐸 (𝑋 ) = 1

(B) 𝐸 (𝑋 ) < 1

(C) 1
𝐸 (𝑋 | 𝑋 > 0) <
2

(D) 1
𝐸 (𝑋 | 𝑋 > 0) >
2

Page 42 of 48

Q.55 Suppose that 𝑈 and 𝑉 are two independent and identically distributed random
variables each having probability density function

𝜆2 𝑥 𝑒 −𝜆𝑥 𝑖𝑓 𝑥 > 0
𝑓 (𝑥 ) = {
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒,

where 𝜆 > 0. Which of the following statements is/are true?

(A) The distribution of 𝑈 − 𝑉 is symmetric about 0

(B) The distribution of 𝑈𝑉 does not depend on 𝜆

(C) 𝑈
The distribution of does not depend on 𝜆
𝑉

(D) 𝑈
The distribution of is symmetric about 1
𝑉

Page 43 of 48

Q.56 Let (𝑋, 𝑌) have joint probability mass function

𝑒 −2
𝑝(𝑥, 𝑦) = { 𝑥! (𝑦 − 𝑥)! 𝑖𝑓 𝑥 = 0, 1, 2, … , 𝑦; 𝑦 = 0, 1, 2, …
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.
Then which of the following statements is/are true?

(A) 𝐸 (𝑋 | 𝑌 = 4) = 2

𝑣
(B) The moment generating function of 𝑌 is 𝑒 2(𝑒 −1) for all 𝑣 ∈ ℝ

(C) 𝐸 (𝑋 ) = 2

(D) −2+(1+𝑒 𝑢 )𝑒 𝑣
The joint moment generating function of (𝑋, 𝑌) is 𝑒 for all
(𝑢, 𝑣 ) ∈ ℝ 2

Page 44 of 48

Q.57 Let {𝑋𝑛 }𝑛≥1 be a sequence of independent and identically distributed random
variables with mean 0 and variance 1, all of them defined on the same
probability space. For 𝑛 = 1, 2, 3, … , let
1
𝑌𝑛 = (𝑋 𝑋 + 𝑋3 𝑋4 + ⋯ + 𝑋2𝑛−1 𝑋2𝑛 ).
𝑛 1 2
Then which of the following statements is/are true?

(A) {√𝑛 𝑌𝑛 }𝑛≥1 converges in distribution to a standard normal random variable

(B) {𝑌𝑛 }𝑛≥1 converges in 2nd mean to 0

(C) 1
{𝑌𝑛 + } converges in probability to 0
𝑛 𝑛≥1

(D) {𝑋𝑛 }𝑛≥1 converges almost surely to 0

Q.58 Consider the following regression model

𝑦𝑡 = 𝛼0 + 𝛼1 𝑡 + 𝛼2 𝑡 2 + 𝜖𝑡 , 𝑡 = 1, 2, … ,100,

where 𝛼0 , 𝛼1 and 𝛼2 are unknown parameters and 𝜖𝑡 ’s are independent and
identically distributed random variables each having 𝑁(𝜇, 1) distribution with
𝜇 ∈ ℝ unknown. Then which of the following statements is/are true?

(A) There exists an unbiased estimator of 𝛼1

(B) There exists an unbiased estimator of 𝛼2

(C) There exists an unbiased estimator of 𝛼0

(D) There exists an unbiased estimator of 𝜇

Page 45 of 48

Q.59 Consider the orthonormal set

1 1
1
√3 √6
1 2 √2
𝑣1 = − , 𝑣2 = , 𝑣3 = 0
√3 √6 1
1 1 −
[ √2]
{ [ √3 ] [ √6 ] }
𝑎
3
with respect to the standard inner product on ℝ . If 𝑢 = [𝑏 ] is the vector such
𝑐
that inner products of 𝑢 with 𝑣1 , 𝑣2 and 𝑣3 are 1, 2 and 3, respectively, then
𝑎2 + 𝑏2 + 𝑐 2 (in integer) equals _______________

Q.60 Consider the probability space (Ω, 𝒢, 𝑃 ), where Ω = {1, 2, 3, 4},
1
𝒢 = {𝜙, Ω, {1}, {4}, {2, 3}, {1, 4}, {1, 2, 3}, {2, 3, 4}}, and 𝑃({1}) = .
4
Let 𝑋 be the random variable defined on the above probability space as
3
𝑋 (1) = 1, 𝑋(2) = 𝑋 (3) = 2 and 𝑋 (4) = 3. If 𝑃(𝑋 ≤ 2) = , then
4
𝑃({1, 4}) (rounded off to two decimal places) equals _______________

Q.61 Let {𝑋𝑛 }𝑛≥1 be a sequence of independent and identically distributed random
variables each having probability density function

𝑒 −𝑥 𝑖𝑓 𝑥 > 0
𝑓 (𝑥 ) = {
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.
1
̅𝑛 =
For 𝑛 ≥ 1, let 𝑌𝑛 = |𝑋2𝑛 − 𝑋2𝑛−1 |. If 𝑌 ∑𝑛𝑖=1 𝑌𝑖 for 𝑛 ≥ 1 and
𝑛
{√𝑛 (𝑒 −𝑌̅𝑛 − 𝑒 −1 )}𝑛≥1 converges in distribution to a normal random variable
with mean 0 and variance 𝜎 2 , then 𝜎 2 (rounded off to two decimal places)

equals _________________

Page 46 of 48

Q.62 Consider a birth-death process on the state space {0, 1, 2, 3}. The birth rates are
given by 𝜆0 = 1, 𝜆1 = 1, 𝜆2 = 2 and 𝜆3 = 0. The death rates are given by
𝜇0 = 0, 𝜇1 = 1, 𝜇2 = 1 and 𝜇3 = 1. If [𝜋0 , 𝜋1 , 𝜋2 , 𝜋3 ] is the unique
stationary distribution, then 𝜋0 + 2𝜋1 + 3𝜋2 + 4𝜋3 (rounded off to two
decimal places) equals _______________

Q.63 1 5
Let {−1, − , 1, , 3} be a realization of a random sample of size 5 from a
2 2
1
population having 𝑁 ( , 𝜎 2 ) distribution, where 𝜎 > 0 is an unknown
2
parameter. Let 𝑇 be an unbiased estimator of 𝜎 2 whose variance attains the
Cramer-Rao lower bound. Then based on the above data, the realized value of 𝑇
(rounded off to two decimal places) equals _______________

Q.64 Let 𝑋 be a random sample of size 1 from a population with cumulative
distribution function

0 𝑖𝑓 𝑥 < 0
𝐹 (𝑥 ) = { 1 − (1 − 𝑥 ) 𝜃 𝑖𝑓 0 ≤ 𝑥 < 1
1 𝑖𝑓 𝑥 ≥ 1,

where 𝜃 > 0 is an unknown parameter. To test 𝐻0 : 𝜃 = 1 against 𝐻1 : 𝜃 = 2,
consider using the critical region {𝑥 ∈ ℝ ∶ 𝑥 < 0.5}. If 𝛼 and 𝛽 denote the
level and power of the test, respectively, then 𝛼 + 𝛽 (rounded off to two
decimal places) equals _______________

Page 47 of 48

Q.65 Let {0.13, 0.12, 0.78, 0.51} be a realization of a random sample of size 4
from a population with cumulative distribution function 𝐹 (⋅). Consider testing

𝐻0 : 𝐹 = 𝐹0 against 𝐻1 : 𝐹 ≠ 𝐹0 ,
where
0 𝑖𝑓 𝑥 < 0
𝐹0 (𝑥) = { 𝑥 𝑖𝑓 0 ≤ 𝑥 < 1
1 𝑖𝑓 𝑥 ≥ 1.

Let 𝐷 denote the Kolmogorov-Smirnov test statistic. If 𝑃 (𝐷 > 0.669) = 0.01
under 𝐻0 and
1 𝑖𝑓 𝐻0 𝑖𝑠 𝑎𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑎𝑡 𝑙𝑒𝑣𝑒𝑙 0.01
𝜓={
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒,
then based on the given data, the observed value of 𝐷 + 𝜓 (rounded off to two
decimal places) equals _______________

END OF QUESTION PAPER

Page 48 of 48
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