GATE 2026 Question Paper Statistics (ST)

FOR GATE EXAM PREPARATION

GATE 2026
Question Paper ·
EXAM YEAR TYPE SUBJECT

GATE 2026 Question Paper Statistics (ST)

Notes · Sample Papers · Previous Year Papers · Mock Tests

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

Q.1 The antonym of the word protagonist is ________.

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(A)
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(B) agantagonist

(C) arsonist

(D) anarchist

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Organizing Institute: IIT Guwahati

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Q.2 The figure shows two 4-tile patterns.

Either one or both of the patterns can be used any number of times and in any
orientation to construct a new pattern. Which one of the options below cannot be
constructed by using only these two 4-tile patterns assuming there are no overlaps
among them?

(A)

(B)

(C)

(D)

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Q.3 Consider a knock-out women’s badminton singles tournament where there are no
ties. The loser in each game is eliminated from the tournament. Every player
plays until she is defeated or remains the last undefeated player. The last
undefeated player is declared the winner of the tournament. If there are 64 players
in the beginning of the tournament, how many games should be played in total to

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declare the winner of the tournament?

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(A)
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(B) 64

(C) 63

(D) 32
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Q.4 A student needs to enroll for a minimum of 60 credits. A student cannot enroll for
more than 70 credits. The credits are divided amongst project and three distinct
sets of courses namely, core courses, specialization courses, and elective courses.
It is compulsory for a student to enroll for exactly 15 credits of core courses and
exactly 20 credits of project. In addition, a student has to enroll for a minimum of
10 credits of specialization courses. The maximum credits of elective courses that
a student can enroll for is ______

(A) 10

(B) 15

(C) 20

(D) 25

Q.5 ‘When the teacher is in the room, all students stand silently.’

If the above statement is true, which one of the following statements is not
necessarily true?

(A) If any student is not standing silently, then the teacher is not in the room.

(B) When the teacher is in the room, all students are silent.

(C) If all students are standing, then the teacher is in the room.

(D) When the teacher is in the room, all students are standing.

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Q.6 – Q.10 Carry TWO marks Each

Q.6 Combinatorics deals with problems involving counting. For example, “How many
distinct arrangements of N distinct objects in M spaces on a circle are possible?”
is a typical problem in combinatorics. This kind of counting is sometimes used in
the modeling of several physical phenomena. Often, in such models, the different
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combinatorial possibilities are assigned probability values. Assigning probabilities

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enables the computation of the average values of physical quantities.
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Consider the following statements:
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Combinatorics is always invoked in the modeling of physical phenomena. ag
P:la
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agQ: Modeling some physical phenomena involves assigning probabilities to
combinatorial possibilities in order to compute average values of physical
quantities.

Based on the passage above, what can be inferred about statements P and Q?

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(A) P is False and Q is False

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(B) P is False and Q is True
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(C) P is True and Q is False

(D) P is True and Q is True

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Q.7 In Panel I of the figure below, the front view and top view of a structure are
shown. Which one of the 3D structures shown in Panel II possesses the views
shown in Panel I?

(A) (i)

(B) (ii)

(C) (iii)

(D) (iv)

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Q.8 For positive real numbers 𝑆 and 𝐾, the function 𝐻𝐾 (𝑆) is defined as:
𝐻𝐾 (𝑆) = max(𝑆 − 𝐾, 0). The max function is defined as:

𝑎, when 𝑎 > 𝑏
max(𝑎, 𝑏) = {
𝑏, when 𝑎 ≤ 𝑏

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The graph below shows the plot of a function 𝑁(𝑆) versus 𝑆.
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𝑁(𝑆) can be expressed as _____.
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(A) 𝐻10 (𝑆) − 𝐻20 (𝑆)

(B) 𝐻10 (𝑆) − 2𝐻20 (𝑆)

(C) −𝐻10 (𝑆) + 𝐻20 (𝑆)
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Organizing Institute: IIT Guwahati

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Q.9 In the 2020 summer Olympics’ Javelin throw finals, Neeraj Chopra exhibited a
spectacular performance to win the gold medal. The silver medal was won by
Jakub Vadlejch and the bronze medal was won by Vitezlav Vesely. There were
six rounds of throws with each athlete having one throw per round. The best of
all the throws of each athlete is considered for the medal. Following were the
observations about the throws:

i. The first and second rounds were dominated by Neeraj Chopra with a gold
medal performance in his second throw, while the other two athletes did
not have any medal winning throws in these rounds.

ii. The throws in the last round by both Jakub Vadlejch and Vitezlav Vesely
were fouls and were not considered for scoring.

iii. After four rounds, Vitezlav Vesely was in the second position and could
not improve upon his best throw in the succeeding rounds.

iv. In the fourth round, the throw by Jakub Vadlejch was the best in that
round.

In which round did Vitezlav Vesely have his best throw?

(A) Third

(B) Fourth

(C) Fifth

(D) Sixth

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Q.10 An unbiased six-faced dice whose faces are marked with numbers 1, 2, 3, 4, 5,
and 6 is rolled twice in succession and the number on the top face is recorded
each time. The probability that the number appearing in the second roll is an
integer multiple of the number appearing in the first roll is __________

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(A)
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(B)
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(C) 7
18

5
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(D)
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Q.11 – Q.35 Carry ONE mark Each

Q.11 Consider the polynomial

𝑝(𝑡) = 𝑡(𝑡 − 1)(𝑡 − 2)

and define
𝑥
1 1
𝐹(𝑥) = ∫ 𝑑𝑡, 𝑥 ∈ [ , 1).
1 𝑝(𝑡) 2
2

Which of the following statements is correct?

(A) 𝐹 is strictly decreasing

(B) 𝐹 is strictly increasing

(C) lim 𝐹(𝑥) exists
𝑥→1

3
(D) 𝐹′ ( ) = 1
4

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Q.12 Let 𝑓: (−1, 1) → ℝ be a differentiable function. Consider the following
statements:

(I) Suppose 𝑓(0) ≥ 0, and 𝑓 ′ (𝑥) > 0 whenever 𝑓(𝑥) = 0, for any 𝑥 ≥ 0.
Then 𝑓(𝑥) > 0, for any 𝑥 > 0.
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(II)
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Suppose 𝑓(0) ≤ 0, and 𝑓 ′ (𝑥) > 0 whenever 𝑓(𝑥) = 0, for any 𝑥 ≤ 0.
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Then
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l of the following statements is correct? ag
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Which

(A) Only statement (I) is correct

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(B) Only statement (II) is correct

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(C) Both statements (I) and (II) are correct

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(D) Neither statement (I) nor statement (II) is correct

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Q.13 Consider the following two subspaces of ℝ4

𝑊1 = {(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) ∶ 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 = 0}

𝑊2 = {(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) ∶ 𝑥1 + 2𝑥2 + 3𝑥3 + 4𝑥4 = 0}.

Which of the following statements is correct?

(A) dim(𝑊1 ∩ 𝑊2 ) = 0

(B) dim(𝑊1 ∩ 𝑊2 ) = 1

(C) dim(𝑊1 ∩ 𝑊2 ) = 2

(D) dim(𝑊1 ∩ 𝑊2 ) = 3

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Q.14 Let 𝑋 and 𝑌 be two independent discrete random variables such that the moment
generating functions of 𝑋 and 𝑋 + 𝑌 are given by

1 + 2𝑒 −𝑡 + 3𝑒 2𝑡
𝑀𝑋 (𝑡) = , 𝑡 ∈ ℝ,
6
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and
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l as
la s 𝑀𝑋+𝑌 (𝑡) =
6
, 𝑡 ∈ ℝ,
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a respectively. Then which of the following statements is correct?

1
(A) 𝑃(𝑋𝑌 = 0) =
6

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(B) 𝑃(𝑋𝑌 = 2) = 6
1
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(C) 𝐸(𝑋) = 0 a
(D) Var(𝑌) = 1

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Organizing Institute: IIT Guwahati

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Q.15 Let 𝑋 and 𝑌 be identically distributed random variables with variance 𝜎 2 ∈ (0, ∞).
Then the correlation coefficient between 𝑋 and 𝑌 is

(A) 𝐸(𝑋−𝑌)2
1−
2 𝜎2

(B) 2 𝐸(𝑋−𝑌)2
1−
𝜎2

(C) 𝐸(𝑋−𝑌)2
1−
𝜎2

(D) 𝐸(𝑋+𝑌)2
1−
𝜎2

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Q.16 Let 𝑋 and 𝑌 be independent and identically distributed normal random variables. If

𝑃(𝑋 + 2𝑌 ≤ 3) = 𝑃(2𝑋 − 𝑌 ≥ 4),

then 𝐸(𝑋) is

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(A)
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(B)
4

(C) 3
7

4
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(D)

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Q.17 Let 𝑋 be a random variable with the following probability density function

2 −2𝑥
𝑓(𝑥) = {4𝑥 𝑒 if 𝑥 > 0
0 otherwise.

If 𝑌 = ln 𝑋, then which of the following statements is correct?

(A) 𝐸(𝑌) is not finite

(B) 3
𝐸(𝑌) = ln
2

(C) 3
𝐸(𝑌) > ln
2

(D) 3
𝐸(𝑌) < ln 2

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Q.18 Let {𝑋𝑛 : 𝑛 ≥ 0} be a homogeneous Markov chain with state space 𝑆 = {1, 2, … , 7}
and transition probability matrix

1 2
0 0 0 0 0
3 3
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3 3 3

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0
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0 0 0 0
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1
0
1
0
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a 1
2 4 4
1 1
0 0 0 0
2 4 4
2 1 1
0 0 0 0
3 6 6
1 1 1
( 3 0 0 0 0
3 3 )

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Then which of the following statements is correct?

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(A) 2 is a transient state
a
(B) 3 is a transient state

(C) 4 is a transient state

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(D) 5 is a transient state
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Q.19 Let 𝑋1 , 𝑋2 be a random sample from the following probability density function

𝛼−1 −𝑥 𝛼
𝑓𝛼 (𝑥) = {𝛼𝑥 𝑒 if 𝑥 > 0
0 otherwise,

where 𝛼 ∈ (0, ∞) is an unknown parameter. Let
𝑋(1) = min{𝑋1 , 𝑋2 } and 𝑋(2) = max{𝑋1 , 𝑋2 }.
Then which of the following statements is correct?

(A) (𝑋1 + 𝑋2 , 𝑋(1) ) is a minimal sufficient statistic

(B) (𝑋1 , 𝑋(2) ) is a minimal sufficient statistic

(C) (𝑋1 − 𝑋2 , 𝑋(1) ) is a sufficient statistic but not a minimal sufficient statistic

(D) (𝑋1 − 𝑋2 , 𝑋(2) ) is a complete sufficient statistic

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Q.20 Let 𝑋1 and 𝑋2 be independent and identically distributed random variables
following normal distribution with mean 𝜃 ∈ (−∞, ∞) and variance 1. Then which
of the following estimators of their expected values attains the Cramer-Rao lower
bound?
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𝑋l + 𝑋
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(A)
a 1 2

(B) 𝑋12 +𝑋22 +5
2

(C) (𝑋1 + 𝑋2 )2

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(D) 2𝑋1 − 𝑋2

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Q.21
a
For testing a null hypothesis 𝐻 against an alternative hypothesis 𝐻 at level of
0 1

significance 𝛼 ∈ (0, 1), which of the following statements is correct?

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(A) A uniformly most powerful test always exists
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s e (B) If a uniformly most powerful test exists then its size is 𝛼
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a (C) If a uniformly most powerful unbiased test exists then it is necessarily a uniformly
most powerful test

(D) If a uniformly most powerful test exists then it is necessarily an unbiased test

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Q.22 Let (𝑋1 , 𝑌1 ), (𝑋2 , 𝑌2 ), … , (𝑋𝑛 , 𝑌𝑛 ), 𝑛 ≥ 2, be a random sample from a continuous
bivariate distribution with joint distribution function 𝐹𝑋,𝑌 . Further, 𝐹𝑋 and 𝐹𝑌 are
the marginal distribution functions of 𝑋 and 𝑌, respectively. If
𝐹𝑋,𝑌 (𝑥, 𝑦) = 𝐹𝑋 (𝑥)𝐹𝑌 (𝑦), ∀ (𝑥, 𝑦),
then, for any two independent pairs (𝑋𝑖 , 𝑌𝑖 ) and (𝑋𝑗 , 𝑌𝑗 ),
𝑃[(𝑋𝑖 − 𝑋𝑗 )(𝑌𝑖 − 𝑌𝑗 ) > 0]
equals

(A) 1
4

(B) 1
2

(C) 3
4

(D) 3
8

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Q.23 For a random sample 𝑋1 , … , 𝑋𝑛 , 𝑛 ≥ 2, from a population with distribution
function 𝐹𝑋 , let 𝑍𝑛 (𝑥) be the proportion of sample values less than or equal to 𝑥,
𝑥 ∈ ℝ. Which of the following statements is/are true?

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(A) 𝐸(𝑍𝑛 (𝑥)) = 𝐹𝑋 (𝑥)
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(B)
a 𝐹𝑋 (𝑥)(1−𝐹𝑋 (𝑥))
Var(𝑍𝑛 (𝑥)) =
𝑛2

(C) Cov(𝑍𝑛 (𝑥), 𝑍𝑛 (𝑦)) = 0, for all 𝑥 ≠ 𝑦

(D) 𝑍𝑛 (𝑥) is a consistent estimator of 𝐹𝑋 (𝑥)
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Q.24 The value of
a 1
1+
2𝑛 2
lim 𝑛 ∫ 𝑒 𝑡 −1 𝑑𝑡
𝑛→∞ 1
1−
2𝑛

equals __________ (answer in integer).

Q.25 Consider the system of linear equations
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s e m
s e 2𝑥 + 𝑦 + 3𝑧 = 6
g l a
g la a
a 3𝑥 + 2𝑦 + 𝑘𝑧 = 𝑘 + 1

If the above system has no solution, then the value of 𝑘 equals _____________
(answer in integer).

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Q.26 Consider the subspace

𝑈 = {(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) ∈ ℝ4 ∶ 𝑥1 + 𝑥2 + 𝑥4 = 0, 𝑥3 = 0}.

Let 𝑈 ⊥ be its orthogonal complement. The value of dim(𝑈 ⊥ ) equals __________
(answer in integer).

Q.27 Let 𝑋 be a random variable such that
𝑃(𝑋 = 𝑖) = 2𝑃(𝑋 = 𝑖 − 1), 𝑖 = 2, 3, … , 𝑛, 𝑛 ≥ 7,
and ∑𝑛𝑖=1 𝑃(𝑋 = 𝑖) = 1. Then (2𝑛 − 1)𝑃(𝑋 = 7) equals ________ (answer in
integer).

Q.28 Let 𝑋 and 𝑌 be two continuous random variables having the following joint
probability density function

𝑥+𝑦 if 0 < 𝑥 < 1, 0 < 𝑦 < 1
𝑓(𝑥, 𝑦) = {
0 otherwise.

Then 72 (Var(𝑋) + Var(𝑌)) equals _____________ (answer in integer).

Q.29 Let 𝑋1 , 𝑋2 and 𝑋3 be three independent random variables such that 𝑋𝑘 (𝑘 = 1, 2, 3)
has the following probability density function

−𝑘𝑥
𝑓𝑘 (𝑥) = {𝑘 𝑒 if 𝑥 > 0
0 otherwise.

Let 𝑌 = min{𝑋1 , 𝑋2 , 𝑋3 }. Then the value of 𝐸(3𝑌 2 − 𝑌) equals ____________
(answer in integer).

Q.30 Let 𝑋 be a continuous random variable having distribution function 𝐹. If
𝑌 = −3 ln 𝐹(𝑋),
then 𝐸(𝑌) equals _______________ (answer in integer).

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Q.31 Let {𝑊(𝑡) ∶ 𝑡 ≥ 0} be a standard Brownian motion, with 𝑊(0) = 0. Define

𝑍1 = 𝑊(1) + 𝑊(2) and 𝑍2 = 𝑊(2) + 𝑊(3).

Let 𝜌 be the correlation coefficient between 𝑍1 and 𝑍2 . Then the value of 10𝜌 is
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__________ (round off to two decimal places).
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Let s𝑥 e, 𝑥 , … , 𝑥 (𝑛 ≥ 2) be the observed values of a random sample from the la
Q.32
l a 1 2 𝑛
a g
agfollowing probability density function
𝜆𝛼 𝛼−1 −𝜆𝑥
𝑥 𝑒 if 𝑥 > 0
𝑓(𝑥) = {Γ(𝛼)
0 otherwise,

where 𝛼 ∈ (0, ∞) and 𝜆 ∈ (0, ∞) are unknown parameters. If

andom
2 2 2
𝑥1 + 𝑥2 + ⋯ + 𝑥𝑛 𝑥 + 𝑥 + ⋯+ 𝑥 1 2 𝑛
𝑛
=2
. c 𝑛
= 5,

s emof 𝛼 equals ________ (answer in integer).
then the method of moments estimate

g la
Q.33
a
Let 𝑋1 , 𝑋2 , … , 𝑋10 be a random sample from the following probability density
function
−(𝑥−𝜇) 2
(𝑥
𝑓(𝑥) = {2 − 𝜇)𝑒 if 𝑥 > 𝜇
0 otherwise,

where 𝜇 ∈ (−∞, ∞) is an unknown parameter. It is given that the observed value
m
m of min{𝑋1 , 𝑋2 , … , 𝑋10 } is 1. Using the pivot min{𝑋1 , 𝑋2 , … , 𝑋10 } − 𝜇, suppose a
.co
m .co 95% confidence interval of 𝜇 is of the form (𝑐, 1), then 𝑐 equals ______________
s e m
s e (rounded off to two decimal places).
g l a
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a

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la
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Q.34 Let 𝑋 be a single observation from a distribution having the probability density
function
1 if 𝜃 < 𝑥 < 𝜃 + 1
𝑓𝜃 (𝑥) = {
0 otherwise,

where 𝜃 ∈ (−∞, ∞). For testing 𝐻0 : 𝜃 ≤ 0 against 𝐻1 : 𝜃 > 1, let 𝛽 be the power
of the uniformly most powerful test of level 0.05. Then 225 𝛽 equals __________
(answer in integer).

Q.35 Let (𝑋1 , 𝑋2 , 𝑋3 )𝑇 follow a trivariate normal distribution with mean vector 𝜇 and
covariance matrix Σ given by
1 2 1 1
𝜇 =(1) and Σ = (1 2 1).
1 1 1 2

Then Var(𝑋1 | 𝑋2 = 1, 𝑋3 = −1) equals ____________ (rounded off to two
decimal places).

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Q.36 – Q.65 Carry TWO marks Each

Q.36 Consider the function 𝑓: ℝ2 → ℝ defined by

𝑓(𝑥1 , 𝑥2 ) = 2𝑥14 + 𝑥22 + 𝑥2 𝑥12 .

m
.co
Which of the following statements is correct?
m
.co s e m
s em l a
g la are two distinct stationary points of 𝑓 ag
(A)
a There

(B) (0, 0) is a saddle point of 𝑓

(C) (0, 0) is a local maximum of 𝑓

(0, 0) is a local minimum of 𝑓
(D)
m
m .co
s e
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a

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a

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.co s e m
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Q.37 Let 𝑥1 ∈ (0, 4) and consider the sequence {𝑥𝑛 }𝑛≥1 defined iteratively by

1
𝑥𝑛+1 = 2 − (4 − 𝑥𝑛 )2 , 𝑛 ≥ 1.

Consider the following statements:

(I) {𝑥𝑛 } converges to 0 .
𝑥 1
(II) { 𝑛+1 } converges to .
𝑥𝑛 4

Which of the following statements is correct?

(A) Only statement (I) is correct

(B) Only statement (II) is correct

(C) Both statements (I) and (II) are correct

(D) Neither statement (I) nor statement (II) is correct

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Q.38 Let 𝐴 ∈ 𝑀𝑛 (ℝ) be an 𝑛 × 𝑛 real matrix, 𝑛 ≥ 2. Consider the following two
statements:

(I) If 𝜆 ∈ ℂ is an eigenvalue of 𝐴, then its complex conjugate 𝜆̅ is also an

m
.co
eigenvalue.

o m
If 𝑣 = (𝑣1 , 𝑣2 , … , 𝑣𝑛 ) ∈ ℂ𝑛 is an eigenvector corresponding to eigenvalue
m
c
(II)
𝜆 =. 𝑥 + 𝑖𝑦, 𝑦 ≠ 0, then e
s emRe(𝑣) = (Re(𝑣 ), … , Re(𝑣 )) and Im(𝑣) = (Im(𝑣 ), … , Im(𝑣 )) glas
g la 1 𝑛
a 1 𝑛

a are linearly independent vectors over ℝ.

Which of the following statements is correct?

m
(A) Only statement (I) is correct
m .co
Only statement (II) is correct as
e
l
ag
(B)

(C) Both statements (I) and (II) are correct

(D) Neither statement (I) nor statement (II) is correct

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a

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Q.39 Consider the system of equations

𝑥 + 2𝑦 − 𝑧 = 𝑎

𝑥 + 𝑦 + 3𝑧 = 𝑏

2𝑥 + 3𝑦 + 2𝑧 = 𝑐

Consider the following statements:

(I) For every (𝑎, 𝑏, 𝑐) ∈ ℝ3 , the above system has a solution.
(II) For (𝑎, 𝑏, 𝑐) = (0, 0, 0), the solution set is given by
{(−7𝑡, 4𝑡, 𝑡) ∶ 𝑡 ∈ ℝ}.

Which of the following statements is correct?

(A) Only statement (I) is correct

(B) Only statement (II) is correct

(C) Both statements (I) and (II) are correct

(D) Neither statement (I) nor statement (II) is correct

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Q.40 Let (Ω, ℱ, 𝑃) be a probability space, where for 𝐴 ⊂ Ω, 𝐴 ≠ 𝜙, 𝐴 ≠ Ω,

1
ℱ = {Ω, 𝜙, 𝐴, 𝐴𝑐 }, and 𝑃(Ω) = 1, 𝑃(𝜙) = 0, 𝑃(𝐴) = = 𝑃(𝐴𝑐 ). Let 𝑋 and 𝑌 be
2

two random variables defined on Ω as follows:
m
m 1 if 𝜔 ∈ 𝐴 1 if 𝜔 ∈ 𝐴𝑐 .co
.co m
𝑋(𝜔) = { and 𝑌(𝜔) = {
e
𝑐
0 if 𝜔 ∈ 𝐴 , 0 if 𝜔 ∈ 𝐴.
e m l as
las
Then which of the following statements is correct?
ag
ag

(A) 𝑋 and 𝑌 have the same distribution function

𝑋 = 𝑌 almost everywhere
m
(B)

m .co
(C) 𝑋 and 𝑌 are independent
s e
g la
(D) 𝐸(𝑋𝑌) = 1 a

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a

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Q.41 Let 𝑋 and 𝑌 be independent and identically distributed geometric random variables
having the following probability mass function

𝑃(𝑋 = 𝑥) = 𝑝(1 − 𝑝)𝑥 , 𝑥 = 0, 1, 2, …,

where 𝑝 ∈ (0 ,1). Then which of the following statements is correct?

(A) 𝑋+𝑌
𝐸(𝑋|𝑋 + 𝑌) =
2

(B) 𝑃(𝑋 = 𝑌) = 1

(C) 1−𝑝
𝑃(𝑋 = 𝑌) =
1+𝑝

(D) 𝑋+𝑌+2
𝐸(𝑋|𝑋 + 𝑌) =
2

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Q.42 Let 𝑋 be a random variable with support 𝑆 = {0, 1, 2, … } and

𝑃(𝑋 ≥ 𝑘 + 1|𝑋 ≥ 𝑘) = 𝑝, 𝑘 ∈ 𝑆, 0 < 𝑝 < 1 .

Then which of the following statements is correct?
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m .co
.co s e m
s em l a
g la ag
(A)
a 𝑃(𝑋 ≥ 𝑘 + 𝑚|𝑋 ≥ 𝑘) = 𝑃(𝑋 ≥ 𝑚), for all 𝑚, 𝑘 ∈ 𝑆

(B) 𝐸(𝑋) > Var(𝑋)

(C) 𝑃(𝑋 ≤ 𝑥) = 1 − 𝑝 𝑥 , for all 𝑥 ∈ 𝑆

𝑃(𝑋 ≤ 𝑘 + 𝑚|𝑋 ≥ 𝑘) = 1 − 𝑝𝑚 , for all 𝑘, 𝑚 ∈ 𝑆
m
.co
(D)

e m
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ag

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Q.43 Let 𝑋1 , 𝑋2 , 𝑋3 be a random sample from a distribution having probability mass
function

𝜃 if 𝑥 = 1
𝑓𝜃 (𝑥) = {1 − 𝜃 if 𝑥 = 2
0, otherwise,

where 𝜃 ∈ Θ = (0, 1). Let 𝑋 = (𝑋1 , 𝑋2 , 𝑋3 ). Then which of the following is NOT
a sufficient statistic for 𝜃?

(A) 𝑇1 (𝑋) = (𝑋1 − 𝑋2 , 𝑋1 + 𝑋2 , 𝑋1 + 𝑋3 )

(B) 𝑇2 (𝑋) = (𝑋1 + 𝑋2 , 𝑋1 − 𝑋3 , 𝑋2 + 𝑋3 )

(C) 𝑇3 (𝑋) = (𝑋1 − 𝑋2 , 𝑋2 − 𝑋3 , 𝑋3 )

(D) 𝑇4 (𝑋) = (𝑋1 + 𝑋2 , 𝑋3 )

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Q.44 Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 (𝑛 ≥ 2) be a random sample from the probability density function
𝑓(𝑥). Consider the following hypotheses:

𝑥 2
1
𝐻0 : 𝑓(𝑥) = 𝑒 − 2 ; −∞ < 𝑥 < ∞
√2 𝜋
m
.co
1 −|𝑥|
m 𝐻1 : 𝑓(𝑥) = 𝑒 ; −∞ < 𝑥 < ∞ .
.co
2
e m
s em 𝐻 against 𝐻 , let 𝑅 denote the critical region based on likelihood ratioglas
For testing
lahaving level 0.05. Then, for some constant 𝑐, the region 𝑅 is
0 1

g a
a
test

(A) {(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) ∶ ∑𝑛𝑖=1(𝑥𝑖 − 1)2 > 𝑐 }

m
.co
(B) {(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) ∶ ∑𝑛𝑖=1(𝑥𝑖 − 1)2 < 𝑐 }

e m
(C)
las
{(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) ∶ ∑𝑛𝑖=1(|𝑥𝑖 | − 1)2 > 𝑐 }

ag
(D) {(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) ∶ ∑𝑛𝑖=1(|𝑥𝑖 | − 1)2 < 𝑐 }

m
m .co
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a

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la
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Q.45 Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 (𝑛 > 1) be a random sample from the following probability
density function

𝛽𝑒 −𝑥 (1 − 𝑒 −𝑥 )𝛽−1 if 𝑥 > 0
𝑓𝛽 (𝑥) = {
0 otherwise,

where 𝛽 > 0 is an unknown parameter. For testing the following hypotheses,

𝐻0 : 𝛽 = 1 against 𝐻1 : 𝛽 > 1,

at level 𝛼 ∈ (0, 1), which of the following statements is correct?

(A) The uniformly most powerful test does not exist

(B) For some constant 𝑎, the critical region of the uniformly most powerful test will
be of the form 𝐶 = {(𝑥1 , 𝑥2 , … , 𝑥𝑛 ): ∑𝑛𝑖=1 𝑥𝑖 > 𝑎}

(C) For some constant 𝑎, the critical region of the uniformly most powerful test will
be of the form 𝐶 = {(𝑥1 , 𝑥2 , … , 𝑥𝑛 ): ∑𝑛𝑖=1 ln(1 − 𝑒 −𝑥𝑖 ) > 𝑎 }

(D) For some constant 𝑎, the critical region of the uniformly most powerful test will
be of the form 𝐶 = {(𝑥1 , 𝑥2 , … , 𝑥𝑛 ): ∑𝑛𝑖=1 ln(1 − 𝑒 −𝑥𝑖 ) < 𝑎 }

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Q.46 Let 𝑋1 , 𝑋2 , … , 𝑋5 be random observations from a continuous distribution. Let 𝜃𝑝 be
𝑝-th population quantile. Consider the following hypotheses

𝐻0 : 𝜃1 = 2.5 against 𝐻1 : 𝜃1 > 2.5.
2 2
m
o m
Let 𝑋(𝑟) denote the 𝑟-th order statistic of the given observations. Then which of the
c m .co
m . a critical region of a level 0.05 test? s e
e
following is
s l a
g l a ag
a
(A) 𝑋(1) > 2.5

(B) 𝑋(2) > 2.5

m
(C) 𝑋(3) > 2.5
m .co
s e
(D) 𝑋(4) > 2.5
g la
a

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Q.47 Suppose that 𝑋 and 𝑌 are independent and identically distributed 𝑁𝑝 (𝜇, Σ) random
vectors, where 𝜇 ∈ ℝ𝑝 and Σ is a positive definite matrix. Let 𝜒𝑚
2
denote chi-square
distribution with 𝑚-degrees of freedom. Then which of the following statements is
correct?

1
(A) (𝑋 − 𝑌)𝑇 Σ −1 (𝑋 − 𝑌) follows 𝜒𝑝2
2

(B) 2 (𝑋 − 𝑌)𝑇 Σ (𝑋 − 𝑌) follows 𝜒𝑝2

1 2
(C) (𝑋 − 𝑌)𝑇 Σ −1 (𝑋 − 𝑌) follows 𝜒2𝑝
2

2
(D) 2 (𝑋 − 𝑌)𝑇 Σ (𝑋 − 𝑌) follows 𝜒2𝑝

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Q.48 Consider the multiple linear regression model

𝑌𝑖 = 𝛽0 + 𝛽1 𝑥𝑖1 + 𝛽2 𝑥𝑖2 + 𝛽3 𝑥𝑖3 + 𝜖𝑖 , 𝑖 = 1, 2, … , 31,

where 𝜖𝑖 are independent and identically distributed 𝑁(0, 1) variables. The 𝐹-test
m
for testing significance of regression rejects the null hypothesis 𝐻0 ∶ 𝛽1 = 𝛽2 =
m .co
.co
𝛽3 = 0 against the alternative 𝐻𝐴 ∶ at least one 𝛽𝑗 is not equal to 0, 𝑗 = 1, 2, 3,
s e m
s em l a
ag
at 5% level of significance. It is given that

g𝐹l
a = 2.96, 𝐹
0.05; 3, 27 = 2.92, 𝐹
a where, for a random variable 𝐹 following
0.05; 3, 30 = 4.01, 𝐹0.025; 3, 27

central 𝐹-distribution with (𝑚, 𝑛)
0.025; 3, 30 = 3.91,

degrees of freedom, 𝑃(𝐹 > 𝐹𝛼; 𝑚,𝑛 ) = 𝛼, 𝛼 ∈ (0, 1). Then the value of 𝑅 2 cannot
be equal to

m
m .co
(A) 0.50
s e
g la
(B) 0.80 a
(C) 0.30

(D) 0.20

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Q.49 Consider the function 𝑓: ℝ2 → ℝ defined by

𝑥3 + 𝑦3
(𝑥, 𝑦) ≠ 0
𝑓(𝑥, 𝑦) = { √𝑥 2 + 2𝑦 2
0 (𝑥, 𝑦) = 0.

Which of the following statements is/are correct?

(A) 𝑓 is continuous at (0, 0)

(B) Partial derivatives 𝑓𝑥 and 𝑓𝑦 exist at (0, 0) and 𝑓𝑥 (0, 0) = 0, 𝑓𝑦 (0, 0) = 0

(C) 𝑓 is not differentiable at (0, 0)

(D) 𝑓 is differentiable

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Q.50 Consider the matrix

𝜆1 1 0
𝐴 = (0 𝜆2 1), 𝜆1 , 𝜆2 , 𝜆3 ∈ ℝ.
0 0 𝜆3

m
.co
Which of the following statements is/are correct?
m
.co s e m
s em l a
g la ag
(A)
a 𝐴 is never diagonalizable

(B) 𝐴 is diagonalizable if and only if 𝜆1 , 𝜆2 , 𝜆3 are distinct

(C) The dimension of eigenspaces corresponding to each 𝜆𝑖 is 1, 𝑖 = 1, 2, 3

m
For any 𝐴, there exist real numbers 𝛼 and 𝛽 such that 𝐴2 + 𝛼𝐴 + 𝛽𝐼3 = 0
.co
(D)

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ag

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Q.51 Consider the real symmetric matrix 𝐴 = (𝑎𝑖𝑗 ) given by

3 1 1
𝐴 = (1 0 2).
1 2 0
Consider the set

3 3
3
𝑆 = {𝑥 = (𝑥1 , 𝑥2 , 𝑥3 ) ∈ ℝ ∶ ∑ ∑ 𝑎𝑖𝑗 𝑥𝑖 𝑥𝑗 = 1 }.
𝑗=1 𝑖=1

Which of the following statements is/are correct?

(A) 𝑆 is empty set

(B) Any line 𝐿 in ℝ3 has at most two points from 𝑆

(C) 𝑆 is bounded

(D) 𝑆 is unbounded

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Q.52 Let 𝑋 be a discrete random variable with support 𝑆 = {1,2,3, … } such that
𝐸(𝑋 2 ) < ∞. Let 𝐹 be the distribution function of 𝑋. Then which of the following
statements is/are correct?

m
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.co s e m
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𝐸(𝑋) = ∑∞
la
(A) 𝑛=1(1 − 𝐹(𝑛 − 1))

g
a 𝐹(𝑋) has discrete uniform distribution
(B)

1
(C) There exists at least one such random variable 𝑋 such that 𝑋 and have the same
𝑋

distribution

m
𝐸(𝑋 2 ) = ∑∞
𝑛=1(2𝑛 − 1)(1 − 𝐹(𝑛 − 1))
.co
(D)

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Q.53 Let {𝑋𝑛 }𝑛≥1 be a sequence of random variables having the following probability
mass function

1 1 𝑥
𝑃(𝑋𝑛 = 𝑥) = (1 − ) , 𝑥 = 0, 1, 2, … ; 𝑛 ∈ ℕ.
5𝑛 5𝑛

𝑋 𝑑
Define 𝑍𝑛 = 𝑛𝑛 , 𝑛 ∈ ℕ, and let 𝑉 be a random variable. If 𝑍𝑛 → 𝑉, as 𝑛 → ∞,

then which of the following statements is/are correct?

(A) 𝑉 has normal distribution with mean 5 and variance 25

(B) 𝑉 has chi-square distribution with 5 degrees of freedom

(C) 𝑉 has exponential distribution with mean 5

𝑉
(D)
e− 5 has uniform distribution over (0, 1)

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Q.54 Let {𝑋𝑘 }𝑘≥1 be a sequence of independent random variables such that

𝑋2𝑘−1 ∼ Bin(1, 𝜃), and 𝑋2𝑘 ∼ Bin(1, 1 − 𝜃), 𝑘 = 1,2,3, …,

where 𝜃 ∈ (0, 1). Let {𝑌𝑘 }𝑘≥1 be another sequence of independent and identically
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c o m
distributed random variables such that 𝑌𝑘 ∼ Poisson(𝜆), 𝜆 > 0. Define, for 𝑛 ∈ ℕ,
m .co
𝑆 =m
. 𝑛 𝑛

s e
a
2 2

s e
2𝑛 ∑(𝑋 − 𝑋 + 1 − 2𝜃) , 𝑊 = ∑ 𝑌
2𝑘−1 2𝑘 and 𝜎 = 2𝑛𝜃(1 − 𝜃).
𝑛 𝑘
g l 2𝑛

l a 𝑘=1 𝑘=1
a
agThen which of the following statements is/are correct?

(A) 𝑛 𝑆2𝑛 𝑑
→ 𝑁(0, 1), as 𝑛 → ∞
𝜎2𝑛 𝑊𝑛
m
m .co
(B) 𝑛 𝑆2𝑛 𝑑
→ 𝑁 (0,
1
s e
) , as 𝑛 → ∞
la
𝜎2𝑛 𝑊𝑛 𝜆2 (1+𝜆)2

ag
(C) 𝑛 𝑆2𝑛 +𝜎2𝑛 𝑊𝑛 𝑑
→ 𝑁(𝜆(1 + 𝜆), 1), as 𝑛 → ∞
𝑛 𝜎2𝑛

(D) 𝑛 𝑆2𝑛 +𝜎2𝑛 𝑊𝑛 𝑑 1
→ 𝑁 (0, ) , as 𝑛 → ∞
𝑛 𝜎2𝑛 𝜆2 (1+𝜆)2

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Q.55 Let {𝑁(𝑡); 𝑡 ≥ 0} be a homogeneous Poisson process with rate 3, and let 𝑇1 denote
the first arrival time. Then which of the following statements is/are correct?

(A) 𝐸(𝑇1 ) = 3

1
(B) 𝐸(𝑇1 ) =
3

6
(C) 𝐸(𝑇1 |𝑁(2) = 4) =
5

3
(D) 𝐸(𝑇1 |𝑁(3) = 4) =
5

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Q.56 Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 (𝑛 ≥ 2) be a random sample from the following probability
density function

1 −|𝑥−𝜇|
𝑓(𝑥) = 𝑒 , − ∞ < 𝑥 < ∞,
2
m
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1
where 𝜇 ∈ (−∞, ∞) is an unknown parameter. Let 𝑋̅ = ∑𝑛𝑖=1 𝑋𝑖 and 𝜇̂ denote the
e
em likelihood estimator of 𝜇, whenever it exists. Then which of the followingglas
𝑛

s
maximum
la is/are correct? a
agstatements

(A) 𝜇̂ will always exist and 𝜇̂ = 𝑋̅

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(B) 𝜇̂ may not always exist

e m
(C)
las
𝜇̂ always exists but it may not be unique

a g
(D) 𝜇̂ is a consistent estimator of 𝜇

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Q.57 Let 𝑋 be a single observation from a distribution having a probability density
function 𝑓𝜃 , 𝜃 ∈ Θ. For testing 𝐻0 ∶ 𝜃 = 1 against 𝐻1 ∶ 𝜃 ≠ 1, under which of the
following options a uniformly most powerful test of level 0.05 exists?

(A) 1
if 0 < 𝑥 < 𝜃
𝑓𝜃 (𝑥) = {𝜃 ; Θ = (0, ∞)
0 otherwise

− (𝑥−𝜃)
𝑓𝜃 (𝑥) = {𝑒 if 𝑥 > 𝜃
(B)
; Θ = (−∞, ∞)
0 otherwise

𝑥
(C) − 1
𝑓 (𝑥) = { 𝑒 if 𝑥 > 0
𝜃
𝜃 𝜃 ; Θ = (0, ∞)
0 otherwise

𝑥−1
(D) −1
𝑓 (𝑥) = {𝜃 𝑒 if 𝑥 > 1
𝜃
𝜃 ; Θ = (0, ∞)
0 otherwise

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𝑇
Q.58 Suppose that 𝑋 = (𝑋1 , … , 𝑋𝑝 ) follows 𝑁𝑝 (𝜇, Σ), where 𝜇 ∈ ℝ𝑝 and Σ is a positive
definite matrix. Let 𝐴 be a 𝑝 × 𝑝 matrix such that 𝐴𝑇 𝐴 = 𝐼𝑝 .

𝑇
Let 𝑌 = (𝑌1 , … , 𝑌𝑝 ) = 𝐴𝑋. Then which of the following statements is/are correct?

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l a ag
𝑇
(A) 𝑌 follows 𝑁 (𝐴𝜇, 𝐴 Σ𝐴) 𝑝

ag
(B) 𝑌 follows 𝑁𝑝 (𝐴𝜇, 𝐴Σ𝐴𝑇 )

Var(∑𝑝𝑖=1 𝑌𝑖 ) = Var(∑𝑖=1 𝑋𝑖 )
𝑝
(C)

(D) ∑𝑝𝑖=1 Var(𝑌𝑖 ) = ∑𝑝𝑖=1 Var(𝑋𝑖 )
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g la
Suppose that 𝑋 = (𝑋 , … , a
Q.59 𝑇
𝑋 ) follows 𝑁 (0, Σ), where Σ is a positive definite
1 𝑝 𝑝

matrix. Then which of the following statements is/are correct?

𝑋1 , … , 𝑋𝑝 are always independent normal random variables m
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(A)
m
c. o(B) s e m
e m 𝑋1 , … , 𝑋𝑝 are normal random variables
l a
las ag
ag (C) 2 2
𝑋 + ⋯ + 𝑋 always follows a chi-square distribution
1 𝑝

(D) Any linear combination of 𝑋1 , … , 𝑋𝑝 is a normal random variable

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Q.60 Let 𝑌 = (𝑌1 , 𝑌2 , 𝑌3 )𝑇 ∼ 𝑁3 (0, 𝐼3 ), where 𝐼3 denotes the identity matrix of order 3.
Let

1 1 1
3 3 3
1 1 1
𝐴= and 𝐵 = 𝐼3 − 𝐴.
3 3 3
1 1 1
(3 3 3)

Let 𝜒12 denote chi-square distribution with 1 degree of freedom. Then which of the
following statements is/are correct?

(A) 𝑌 𝑇 𝐴𝑌 ∼ 𝜒12

(B) 𝑌 𝑇 𝐵𝑌 ∼ 𝜒12

(C) 𝑌 𝑇 𝐴𝑌 and 𝑌1 − 2𝑌2 + 𝑌3 are independently distributed

(D) 𝑌 𝑇 𝐴𝑌 and 𝑌1 + 2𝑌2 + 𝑌3 are independently distributed

Let 𝐷 = {(𝑥, 𝑦) ∈ ℝ2 : 0 < 𝑥 2 + 𝑦 2 ≤ 1}. For 𝛼 ≥ 0, consider the integral
Q.61
2
1
𝐼𝛼 = ∬ 𝑑𝑥 𝑑𝑦.
𝐷 (𝑥 2 + 𝑦 2 )𝛼

Let

𝑁0 = sup{𝛼 ≥ 0 ∶ 𝐼𝛼 < ∞}.

Then the value of 𝑁0 equals _______________ (answer in integer).

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Q.62 Let 𝑋 follow 𝑁(3, 1). Then the value of 𝐸(𝑋 4 (𝑋 − 3)) equals ______________
(answer in integer).

Q.63 Let 𝑈1 , 𝑈2 , 𝑈3 be three independent exponential random variables such that 𝑈𝑘 has
m
m
the following probability density function
.co
m .co s e m
s e 𝑓𝑘 (𝑥) = {
𝑘 𝑒 −𝑘𝑥 if 𝑥 > 0
𝑘 = 1, 2, 3.
l a
a ag
0 otherwise,
g l
a Let
𝑋 = min{𝑈1 , 𝑈3 } and 𝑌 = min{𝑈2 , 𝑈3 } .
Then 𝑃(𝑋 = 𝑌) equals ______________ (rounded off to two decimal places).

Q.64 Let 𝑋1 , 𝑋2 be a random sample from a distribution having the population density

m
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function

1m
e
𝑓(𝑥) a
l s
= {𝜃 if 0 < 𝑥 < 𝜃

ag 0 otherwise,
where 𝜃 ∈ (0, ∞). Let 𝑋(2) = max{𝑋1 , 𝑋2 } and
𝜓(𝜃) = 𝑃𝜃 (𝑋1 + 𝑋2 < 1), 𝜃 > 0.
Let 𝛿(𝑋(2) ) be an unbiased estimator of 𝜓(𝜃) that depends on observations 𝑋1 and
𝑋2 only through 𝑋(2) . If 𝛿(𝑡) is a continuous function on (0, ∞), then the value of

m
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3
18 𝛿 ( ) equals ______________ (answer in integer).
m 4

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Q.65 Let 𝑋 be a single observation from a distribution having probability density
function

2𝑥
𝑓𝜃 (𝑥) = { 𝜃 2 if 0 < 𝑥 < 𝜃
0 otherwise,

where 𝜃 ∈ (0, ∞). For testing 𝐻0 : 𝜃 ≤ 1 against 𝐻1 : 𝜃 > 1, at level of
significance 0.05, let 𝛽1 be the size of the uniformly most powerful test and 𝛽2 be
the power of the uniformly most powerful test at 𝜃 = 2. Then 10 𝛽1 + 40 𝛽2 equals
______________ (answer in integer).

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