GATE 2025 Question Paper Statistics (ST)

GATE
2025
Question Paper | Answer Key
Graduate Aptitude Test in Engineering
(GATE) is a prestigious national-level exam
that assesses candidates for
comprehensive understanding in various
undergraduate-level subjects in
Engineering, Technology, Science,
Architecture, and Humanities.

General Aptitude
Q.1 – Q.5 Carry ONE mark Each

Q.1 Even though I had planned to go skiing with my friends, I had to __________ at the
last moment because of an injury.

Select the most appropriate option to complete the above sentence.

(A) back up

(B) back of

(C) back on

(D) back out

Q.2 The President, along with the Council of Ministers, ___________ to visit India next
week.

Select the most appropriate option to complete the above sentence.

(A) wish

(B) wishes

(C) will wish

(D) is wishing

Page 1 of 36

Q.3 An electricity utility company charges ₹ 7 per kWh (kilo watt-hour). If a 40-watt
desk light is left on for 10 hours each night for 180 days, what would be the cost of
energy consumption? If the desk light is on for 2 more hours each night for the 180
days, what would be the percentage-increase in the cost of energy consumption?

(A) ₹ 604.8; 10%

(B) ₹ 504; 20%

(C) ₹ 604.8; 12%

(D) ₹ 720; 15%

Page 2 of 36

Q.4 In the context of the given figure, which one of the following options correctly
represents the entries in the blocks labelled (i), (ii), (iii), and (iv), respectively?

N U F (i)

21 14 9 6

H L (ii) O

12 (iv) 15 (iii)

(A) Q, M, 12, and 8

(B) K, L, 10 and 14

(C) I, J, 10, and 8

(D) L, K, 12 and 8

Page 3 of 36

Q.5 A bag contains Violet (V), Yellow (Y), Red (R), and Green (G) balls. On counting
them, the following results are obtained:

(i) The sum of Yellow balls and twice the number of Violet balls is 50.
(ii) The sum of Violet and Green balls is 50.
(iii) The sum of Yellow and Red balls is 50.
(iv) The sum of Violet and twice the number of Red balls is 50.

Which one of the following Pie charts correctly represents the balls in the bag?

(A)
V
10%

G R
40% 20%

Y
30%

(B)

G
30% V
40%

Y
20% R
10%

Page 4 of 36

(C)

G
20% V
30%
Y
10%

R
40%

(D)
G
10% V
20%

Y
40% R
30%

Page 5 of 36

Q.6 – Q.10 Carry TWO marks Each

Q.6 “His life was divided between the books, his friends, and long walks. A solitary
man, he worked at all hours without much method, and probably courted his fatal
illness in this way. To his own name there is not much to show; but such was his
liberality that he was continually helping others, and fruits of his erudition are
widely scattered, and have gone to increase many a comparative stranger’s
reputation.”

(From E.V. Lucas’s “A Funeral”)

Based only on the information provided in the above passage, which one of the
following statements is true?

(A) The solitary man described in the passage is dead.

(B) Strangers helped create a grand reputation for the solitary man described in the
passage.

(C) The solitary man described in the passage found joy in scattering fruits.

(D) The solitary man worked in a court where he fell ill.

Page 6 of 36

Q.7 For the clock shown in the figure, if

O* = O Q S Z P R T, and

X* = X Z P W Y O Q ,

then which one among the given options is most appropriate for P* ?

(A) P U W R T V X

(B) P R T O Q S U

(C) P T V Q S U W

(D) P S U P R T V

Page 7 of 36

Q.8 Consider a five-digit number 𝑃𝑄𝑅𝑆𝑇 that has distinct digits 𝑃, 𝑄, 𝑅, 𝑆, and 𝑇, and
satisfies the following conditions:

𝑃<𝑄

𝑆>𝑃>𝑇

𝑅<𝑇

If integers 1 through 5 are used to construct such a number, the value of 𝑃 is:

(A) 1

(B) 2

(C) 3

(D) 4

Q.9 A business person buys potatoes of two different varieties P and Q, mixes them in
a certain ratio and sells them at ₹ 192 per kg.

The cost of the variety P is ₹ 800 for 5 kg.

The cost of the variety Q is ₹ 800 for 4 kg.

If the person gets 8% profit, what is the P:Q ratio (by weight)?

(A) 5:4

(B) 3:4

(C) 3:2

(D) 1:1

Page 8 of 36

Q.10 Three villages P, Q, and R are located in such a way that the distance PQ = 13 km,
QR = 14 km, and RP = 15 km, as shown in the figure. A straight road joins Q and
R. It is proposed to connect P to this road QR by constructing another road. What
is the minimum possible length (in km) of this connecting road?

Note: The figure shown is representative.

P

13 km 15 km

Q R
14 km

(A) 10.5

(B) 11.0

(C) 12.0

(D) 12.5

Page 9 of 36

Q.11 – Q.35 Carry ONE mark Each

Q.11 Let 𝑓: [0, ∞) → [0, ∞) be a differentiable function with 𝑓(𝑥) > 0 for all 𝑥 > 0,
and 𝑓(0) = 0. Further, 𝑓 satisfies
𝑥
2
(𝑓(𝑥))2 = ∫ ((𝑓(𝑡)) + 𝑓(𝑡)) 𝑑𝑡 , 𝑥 > 0.
0

Then which one of the following options is correct?

(A) 0 < 𝑓(2) ≤ 1

(B) 1 < 𝑓(2) ≤ 2

(C) 2 < 𝑓(2) ≤ 3

(D) 3 < 𝑓(2) ≤ 4

Q.12 Among the following four statements about countability and uncountability of
different sets, which is the correct statement?

𝑛
(A) The set ⋃∞ 𝑖
𝑛=0{𝑥 ∈ ℝ: 𝑥 = ∑𝑖=0 10 𝑎𝑖 , where 𝑎𝑖 ∈ {1, 2} for 𝑖 = 0, 1, 2, … , 𝑛} is
uncountable

(B) 𝑎
The set {𝑥 ∈ (0,1): 𝑥 = ∑∞ 𝑛
𝑛=1 10𝑛 , where 𝑎𝑛 = 1 or 2 for each 𝑛 ∈ ℕ} is
uncountable

(C) There exists an uncountable set whose elements are pairwise disjoint open
intervals in ℝ

(D) The set of all intervals with rational end points is uncountable

Page 10 of 36

Q.13 Let 𝑆 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 \{(0,0,0)} ∶ 𝑧 = −(𝑥 + 𝑦) }. Denote

𝑆 ⊥ = {(𝑝, 𝑞, 𝑟) ∈ ℝ3 ∶ 𝑝𝑥 + 𝑞𝑦 + 𝑟𝑧 = 0 for all (𝑥, 𝑦, 𝑧) ∈ 𝑆}.

Then which one of the following options is correct?

(A) 𝑆 ⊥ is not a subspace of ℝ3

(B) 𝑆 ⊥ = {(0,0,0)}

(C) dim(𝑆 ⊥ ) = 1

(D) dim(𝑆 ⊥ ) = 2

Q.14 Let 𝑋 be a random variable having the Poisson distribution with mean log 𝑒 2. Then
E(𝑒 (log𝑒 3)𝑋 ) equals

(A) 1

(B) 2

(C) 3

(D) 4

Page 11 of 36

Q.15 Let (𝑋1 , 𝑋2 , 𝑋3 ) follow the multinomial distribution with the number of trials being
3 1 3
100 and the probability vector (10 , 10 , 5). Then E(X2 | 𝑋3 = 40) equals

(A) 25

(B) 15

(C) 30

(D) 45

Q.16 Let {𝑋𝑛 }𝑛≥1 be a sequence of i.i.d. random variables with the common probability
density function

1
𝑓(𝑥) = , −∞<𝑥 <∞.
𝜋(1 + 𝑥 2 )
1 1
Define 𝑌𝑛 = 2 + π tan−1(𝑋𝑛 ) for 𝑛 = 1, 2, … . Then which one of the following
options is correct?

(A) 1 𝑃 1
∑𝑛𝑖=1 𝑌𝑖 → as 𝑛 → ∞
𝑛 2

(B) 1 𝑃
∑𝑛𝑖=1 𝑌𝑖 → 0 as 𝑛 → ∞
𝑛

(C) 1 𝑃
∑𝑛𝑖=1 𝑋𝑖 → 0 as 𝑛 → ∞
𝑛

(D) 1 𝑃 1
∑𝑛𝑖=1 𝑋𝑖 → as 𝑛 → ∞
𝑛 2

Page 12 of 36

Q.17 Let {𝑁(𝑡): 𝑡 ≥ 0} be a homogenous Poisson process with the intensity/rate 𝜆 = 2.
Let

𝑋 = 𝑁(6) − 𝑁(1)
𝑌 = 𝑁(5) − 𝑁(3)
𝑊 = 𝑁(6) − 𝑁(5)
𝑍 = 𝑁(3) − 𝑁(1).

Then which one of the following options is correct?

(A) Cov(𝑊, 𝑍) = 2

(B) 𝑌 + 𝑍 ∼ Poisson(10)

(C) Pr(𝑌 = 𝑍) = 1

(D) Cov(𝑋, 𝑌) = 4

Q.18 Let 𝑇 be a complete and sufficient statistic for a family 𝒫 of distributions and let 𝑈
be a sufficient statistic for 𝒫. If 𝑃𝑓 (𝑇 ≥ 0) = 1 for all 𝑓 ∈ 𝒫, then which one of the
following options is NOT necessarily correct?

(A) 𝑇 2 is a complete statistic for 𝒫

(B) 𝑇 2 is a minimal sufficient statistic for 𝒫

(C) 𝑇 is a function of 𝑈

(D) 𝑈 is a function of 𝑇

Page 13 of 36

Q.19 Let 𝑋1 , 𝑋2 be a random sample from 𝑁(𝜃, 1) distribution, where 𝜃 ∈ ℝ. Consider
testing 𝐻0 : 𝜃 = 0 against 𝐻1 : 𝜃 ≠ 0. Let 𝜙(𝑋1 , 𝑋2 ) be the likelihood ratio test of
size 0.05 for testing 𝐻0 against 𝐻1 . Then which one of the following options is
correct?

(A) 𝜙(𝑋1 , 𝑋2 ) is a uniformly most powerful test of size 0.05

(B) E𝜃 (𝜙(𝑋1 , 𝑋2 )) ≥ 0.05 ∀ 𝜃 ∈ ℝ

(C) There exists a uniformly most powerful test of size 0.05

(D) E𝜃=0 (𝑋1 𝜙(𝑋1 , 𝑋2 )) = 0.05

Q.20 Let a random variable 𝑋 follow a distribution with density 𝑓 ∈ {𝑓0 , 𝑓1 }, where

1 if 0 ≤ 𝑥 ≤ 1 1 if 1 ≤ 𝑥 ≤ 2
𝑓0 (𝑥) = { and 𝑓1 (𝑥) = {
0 otherwise 0 otherwise.
Let 𝜙 be a most powerful test of level 0.05 for testing 𝐻0 : 𝑓 = 𝑓0 against
𝐻1 : 𝑓 = 𝑓1 based on 𝑋. Then which one of the following options is necessarily
correct?

(A) E𝑓0 (𝜙(𝑋)) = 0.05

(B) E𝑓1 (𝜙(𝑋)) = 1

(C) 𝑃𝑓 (𝜙(𝑋) = 1) = 𝑃𝑓 (𝑋 > 1), ∀ 𝑓 ∈ {𝑓0 , 𝑓1 }

(D) 𝑃𝑓1 (𝜙(𝑋) = 1) < 1

Page 14 of 36

Q.21 Let 𝑋 be a random variable having probability density function 𝑓 ∈ {𝑓0 , 𝑓1 }. Let 𝜙
be a most powerful test of level 0.05 for testing 𝐻0 : 𝑓 = 𝑓0 against 𝐻1 : 𝑓 = 𝑓1 based
on 𝑋. Then which one of the following options is NOT necessarily correct?

(A) 𝜙 is the unique most powerful test of level 0.05

(B) E𝑓1 (𝜙(𝑋)) ≥ 0.05

(C) E𝑓0 (𝜙(𝑋)) ≤ 0.05

(D) For some constant 𝑐 ≥ 0, 𝑃𝑓 (𝑓1 (𝑋) > 𝑐𝑓0 (𝑋)) ≤ 𝑃𝑓 (𝜙(𝑋) = 1), ∀ 𝑓 ∈ {𝑓0 , 𝑓1 }

Q.22 Let {𝑋𝑛 }𝑛≥1 be a sequence of i.i.d. random variables with common distribution
function 𝐹, and let 𝐹𝑛 be the empirical distribution function based on
{𝑋1 , 𝑋2 , … , 𝑋𝑛 }. Then, for each fixed 𝑥 ∈ (−∞, ∞), which one of the following
options is correct?

(A) 𝑃
√𝑛(𝐹𝑛 (𝑥) − 𝐹(𝑥)) → 0 as 𝑛 → ∞

(B) 𝑛(𝐹𝑛 (𝑥) − 𝐹(𝑥)) 𝑑
→ 𝑍 as 𝑛 → ∞, where 𝑍 ∼ 𝑁(0,1)
√𝐹(𝑥)(1 − 𝐹(𝑥))

𝑎.𝑠.
(C) 𝐹𝑛 (𝑥) → 𝐹(𝑥) as 𝑛 → ∞

(D) lim 𝑛 Var(𝐹𝑛 (𝑥)) = 0
𝑛→∞

Page 15 of 36

Q.23 Let (𝑋, 𝑌)𝑇 follow a bivariate normal distribution with E(𝑋) = 3, E(𝑌) = 4,
Var(𝑋) = 25, Var(𝑌) = 100, and Cov(𝑋, 𝑌) = 50𝜌, where 𝜌 ∈ (−1,1). If
E(𝑌 | 𝑋 = 5) = 4.32, then 𝜌 equals

(A) 0.08

(B) 0.8

(C) 0.32

(D) 0.5

Q.24 For a given data (𝑥𝑖 , 𝑦𝑖 ), 𝑖 = 1, 2, … , 𝑛, with ∑𝑛𝑖=1 𝑥𝑖2 > 0, let 𝛽̂ satisfy
𝑛 𝑛
2
∑(𝑦𝑖 − 𝛽̂ 𝑥𝑖 ) = inf ∑(𝑦𝑖 − 𝛽𝑥𝑖 )2 .
𝛽∈ℝ
𝑖=1 𝑖=1

Further, let 𝑣𝑗 = 𝑦𝑗 − 𝑥𝑗 and 𝑢𝑗 = 2𝑥𝑗 , for 𝑗 = 1,2, … , 𝑛, and let 𝛾̂ satisfy
𝑛 𝑛

∑(𝑣𝑖 − 𝛾̂𝑢𝑖 )2 = inf ∑(𝑣𝑖 − 𝛾𝑢𝑖 )2 .
𝛾∈ℝ
𝑖=1 𝑖=1

If 𝛽̂ = 10, then the value of 𝛾̂ is

(A) 4.5

(B) 5

(C) 10

(D) 9

Page 16 of 36

Q.25 Let
1 1
𝐼 = 𝜋 2 ∫ ∫ 𝑦 2 cos 𝜋(1 + 𝑥𝑦) 𝑑𝑥 𝑑𝑦 .
0 0

The value of 𝐼 is equal to ___________ (answer in integer).

Q.26 1 2
Let 𝑃 = ( ) and 𝑄 = 𝑃3 − 2𝑃2 − 4𝑃 + 13𝐼2 , where 𝐼2 denotes the identity
−1 4
matrix of order 2. Then the determinant of 𝑄 is equal to ____________ (answer in
integer).

Q.27 Let 𝑇: ℝ3 → ℝ3 be a linear map defined by

𝑇(𝑥1 , 𝑥2 , 𝑥3 ) = (3𝑥1 + 5𝑥2 + 𝑥3 , 𝑥3 , 2𝑥1 + 2𝑥3 ).

Then the rank of 𝑇 is equal to ____________ (answer in integer).

Q.28 Let 𝑋 be a random variable with distribution function 𝐹, such that

1 3
lim− 𝐹(3 + ℎ) = and 𝐹(3) = .
ℎ→0 4 4

Then 16 Pr(𝑋 = 3) equals ____________ (answer in integer).

Page 17 of 36

Q.29 1
Let 𝑋 ∼ 𝐵𝑖𝑛 (2, 3). Then 18 E(𝑋 2 ) equals ____________ (answer in integer).

Q.30 Let 𝑋 follow a 10-dimensional multivariate normal distribution with zero mean
vector and identity covariance matrix. Define 𝑌 = log 𝑒 √𝑋 𝑇 𝑋 and let 𝑀𝑌 (𝑡)
denote the moment generating function of 𝑌 at 𝑡, 𝑡 > −10. Then 𝑀𝑌 (2) equals
____________ (answer in integer).

Q.31 2
Let {𝑊(𝑡): 𝑡 ≥ 0} be a standard Brownian motion. Then E ((𝑊(2) + 𝑊(3)) )
equals ____________ (answer in integer).

Q.32 Let 𝑥1 = 0, 𝑥2 = 1, 𝑥3 = 1, 𝑥4 = 1 and 𝑥5 = 0 be observed values of a random
sample of size 5 from 𝐵𝑖𝑛(1, 𝜃) distribution, where 𝜃 ∈ (0, 0.7]. Then the
maximum likelihood estimate of 𝜃 based on the above sample is ____________
(rounded off to two decimal places).

Q.33 Let 𝑋1 , … , 𝑋5 be a random sample from 𝑁(𝜃, 6), where 𝜃 ∈ ℝ, and let 𝑐(𝜃) be the
Cramer-Rao lower bound for the variances of unbiased estimators of 𝜃 based on the
above sample. Then 15 inf 𝑐(𝜃) equals ____________ (answer in integer).
𝜃∈ℝ

Page 18 of 36

Q.34 Let (1, 3), (2, 4), (7, 8) be three independent observations. Then the sample
Spearman rank correlation coefficient based on the above observations is
____________ (rounded off to two decimal places).

Q.35 Consider the multi-linear regression model

𝑦𝑖 = 𝛽0 + 𝛽1 𝑥1𝑖 + 𝛽2 𝑥2𝑖 + 𝛽3 𝑥3𝑖 + 𝛽4 𝑥4𝑖 + 𝜖𝑖 , 𝑖 = 1, 2, … , 25 ,

where 𝛽𝑖 , 𝑖 = 0, 1, 2, 3, 4, are unknown parameters, the errors 𝜖𝑖′ 𝑠 are i.i.d. random
variables having 𝑁(0, 𝜎 2 ) distribution, where 𝜎 > 0 is unknown. Suppose that the
5
value of the coefficient of determination 𝑅 2 is obtained as . Then the value of
6
adjusted 𝑅 2 is __________ (rounded off to two decimal places).

Page 19 of 36

Q.36 – Q.65 Carry TWO marks Each

Q.36 Let ℱ = {𝑓: [𝑎, 𝑏] → ℝ | 𝑓 is continuous on [a, b] and differentiable on (𝑎, 𝑏)}.
Which one of the following options is correct?

(A) There exists a non-constant 𝑓 ∈ ℱ such that |𝑓(𝑥) − 𝑓(𝑦)| ≤ |𝑥 − 𝑦|2 for all
𝑥, 𝑦 ∈ [𝑎, 𝑏]

(B) If 𝑓 ∈ ℱ and 𝑥0 ∈ (𝑎, 𝑏), then there exist distinct 𝑥1 , 𝑥2 ∈ [𝑎, 𝑏] such that
𝑓(𝑥1 )−𝑓(𝑥2 )
= 𝑓 ′ (𝑥0 )
𝑥1 −𝑥2

(C) Let 𝑓 ∈ ℱ and 𝑓 ′ (𝑥) ≥ 0 for all 𝑥 ∈ (𝑎, 𝑏). If 𝑓 ′ is zero only at two distinct points,
then 𝑓 is strictly increasing

(D) Let 𝑓 ∈ ℱ. If 𝑓 ′ (𝑥1 ) < 𝑐 < 𝑓 ′ (𝑥2 ) for some 𝑥1 , 𝑥2 ∈ (𝑎, 𝑏), then there may NOT
exist an 𝑥0 ∈ (𝑥1 , 𝑥2 ) such that 𝑓 ′ (𝑥0 ) = 𝑐

Q.37 Let 𝑈 = {(𝑥, 𝑦) ∈ ℝ2 : 𝑥 + 𝑦 ≤ 2}. Define 𝑓: 𝑈 → ℝ by

𝑓(𝑥, 𝑦) = (𝑥 − 1)4 + (𝑦 − 2)4 .

The minimum value of 𝑓 over 𝑈 is

(A) 0

(B) 1
16

(C) 17
81

(D) 1
8

Page 20 of 36

Q.38 Let 𝑃 = (𝑎𝑖𝑗 ) be a 10 × 10 matrix with

1
− if 𝑖 ≠ 𝑗
𝑎𝑖𝑗 = { 10
9
if 𝑖 = 𝑗.
10
Then rank(𝑃) equals

(A) 10

(B) 9

(C) 1

(D) 8

Q.39 Let 𝑋 be a random variable with the distribution function

0 if 𝑥 < 0
𝐹(𝑥) = { 𝛼(1 + 2𝑥 2 ) if 0 ≤ 𝑥 < 1
1 if 𝑥 ≥ 1,
1
where 𝛼 is a real constant. If the median of 𝑋 is , then the value of 𝛼 equals
√2

(A) 1
2

(B) 1
3

(C) 1
4

(D) 1
6

Page 21 of 36

Q.40 Let 𝑋 be a continuous random variable with probability density function

1 1 log 𝑥−𝜇 2
− ( e )
𝑓(𝑥) = 𝑒 2 𝜎 , 𝑥 > 0,
𝜎𝑥 √2𝜋
E(𝑋 2 )
where 𝜇 ∈ ℝ, 𝜎 > 0. If log e ( 2 ) = 4, then Var(log e 𝑋) equals
(E(𝑋))

(A) 2

(B) 4

(C) 16

(D) 64

Q.41 Let 𝑋 and 𝑌 be discrete random variables with joint probability mass function

𝜆𝑛 𝑒 −𝜆
𝑝𝑋,𝑌 (𝑚, 𝑛) = , 𝑚 = 0, … , 𝑛, and 𝑛 = 0, 1, 2, …,
2𝑛 𝑚! (𝑛 − 𝑚)!

where 𝜆 is a fixed positive real number. Then which one of the following options is
correct?

(A) The marginal distribution of 𝑋 is Poisson with mean 𝜆

(B) The marginal distribution of 𝑌 is Poisson with mean 2𝜆

(C) 1
The conditional distribution of 𝑋 given 𝑌 = 3 is 𝐵𝑖𝑛 (3, 2)

(D) 𝜆
E(𝑌 | 𝑋 = 2) = 2

Page 22 of 36

Q.42 Let 𝑋1 , … , 𝑋𝑛 , 𝑛 ≥ 2, be a random sample from a 𝑁(−𝜃, 𝜃) distribution, where
𝜃 > 0 is an unknown parameter. Then which one of the following options is
correct?

(A) ∑𝑛𝑖=1 𝑋𝑖 is a minimal sufficient statistic

(B) ∑𝑛𝑖=1 𝑋𝑖 2 is a minimal sufficient statistic

(C) 1 1 1 2
( ∑𝑛𝑖=1 𝑋𝑖 , ∑𝑛𝑗=1 (𝑋𝑗 − ∑𝑛𝑖=1 𝑋𝑖 ) ) is a complete statistic
𝑛 𝑛−1 𝑛

1
(D) − 𝑛 ∑𝑛𝑖=1 𝑋𝑖 is a uniformly minimum variance unbiased estimator of 𝜃

Page 23 of 36

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

1 −𝑥
𝑓𝜃 (𝑥) = {𝜃 𝑒 if 𝑥 > 0
𝜃

0 otherwise,

where 𝜃 ∈ (0, ∞) is an unknown parameter. For testing 𝐻0 : 𝜃 ≤ 1 against
𝐻1 : 𝜃 > 1, consider the test

1 if 𝑋1 > 1
𝜙(𝑋1 , 𝑋2 ) = {
0 otherwise.
Then which one of the following tests has the same power function as 𝜙?

(A) 𝑋1 +𝑋2 −1
if 𝑋1 + 𝑋2 > 1
𝜙1 (𝑋1 , 𝑋2 ) = { 𝑋1+𝑋2
0 otherwise

(B) 2𝑋1 +2𝑋2 −1
if 𝑋1 + 𝑋2 > 1
𝜙2 (𝑋1 , 𝑋2 ) = { 2(𝑋1 +𝑋2)
0 otherwise

(C) 3𝑋1 +3𝑋2 −1
if 𝑋1 + 𝑋2 > 1
𝜙3 (𝑋1 , 𝑋2 ) = { 3(𝑋1 +𝑋2)
0 otherwise

(D) 4𝑋1 +4𝑋2 −1
if 𝑋1 + 𝑋2 > 1
𝜙4 (𝑋1 , 𝑋2 ) = { 4(𝑋1+𝑋2)
0 otherwise

Page 24 of 36

Q.44 Let 𝑋, 𝑌1 , 𝑌2 be independent random variables such that 𝑋 has the probability
density function

2𝑒 −2𝑥 if 𝑥 ≥ 0
𝑓(𝑥) = {
0 otherwise,

and 𝑌1 and 𝑌2 are identically distributed with probability density function

𝑒 −𝑥 if 𝑥 ≥ 0
𝑔(𝑥) = {
0 otherwise.
For 𝑖 = 1, 2, let 𝑅𝑖 denote the rank of 𝑌𝑖 among 𝑋, 𝑌1 , 𝑌2 . Then E(𝑅1 + 𝑅2 ) equals

(A) 13
3

(B) 22
5

(C) 21
5

(D) 9
2

Page 25 of 36

Q.45 Let 𝑋1 , 𝑋2 , … , 𝑋5 be i.i.d. random vectors following the bivariate normal
distribution with zero mean vector and identity covariance matrix. Define 5 × 2
matrix 𝑋 as 𝑋 = (𝑋1 , 𝑋2 , … , 𝑋5 )𝑇 . Further, let 𝑊 = (𝑊𝑖𝑗 ) = 𝑋 𝑇 𝑋, and
𝑍 = 𝑊11 + 4𝑊12 + 4𝑊22 . Then Var(𝑍) equals

(A) 150

(B) 200

(C) 250

(D) 300

Page 26 of 36

Q.46 Consider the simple linear regression model

𝑦𝑖 = 𝛼 + 𝛽𝑥𝑖 + 𝜖𝑖 , 𝑖 = 1, 2, … ,24,

where 𝛼 ∈ ℝ and 𝛽 ∈ ℝ are unknown parameters, the errors 𝜖𝑖 ′𝑠 are i.i.d. random
variables having 𝑁(0, 𝜎 2 ) distribution, where 𝜎 > 0 is unknown. Suppose the
following summary statistics are obtained from a data set of 24 observations
(𝑥1 , 𝑦1 ), … , (𝑥24 , 𝑦24 ):
24 24

𝑆𝑥𝑥 = ∑(𝑥𝑖 − 𝑥̅ )2 = 22.82 , 𝑆𝑦𝑦 = ∑(𝑦𝑖 − 𝑦̅)2 = 43.62 ,
𝑖=1 𝑖=1

24

and 𝑆𝑥𝑦 = ∑(𝑥𝑖 − 𝑥̅ )(𝑦𝑖 − 𝑦̅) = 15.48 ,
𝑖=1

1 1
where 𝑥̅ = 24 ∑24 𝑖=1 𝑥𝑖 and 𝑦̅ = 24 ∑24𝑖=1 𝑦𝑖 . Then, for testing 𝐻0 : 𝛽 = 0 against
𝐻1 : 𝛽 ≠ 0, the value of the 𝐹- test statistic based on the least squares estimator of
𝛽, whose distribution is 𝐹1,22 , equals (rounded off to two decimal places)

(A) 2.54

(B) 2.98

(C) 3.17

(D) 6.98

Page 27 of 36

Q.47 1 1 1
Let {𝑥𝑛 }𝑛≥1 be a sequence defined as 𝑥𝑛 = 1 + + + ⋯ + 𝑛 − 2(√𝑛 − 1).
√2 √3 √

Then which of the following options is/are correct?

(A) The sequence {𝑥𝑛 }𝑛≥1 is unbounded

(B) The sequence {𝑥𝑛 }𝑛≥1 is monotonically decreasing

(C) The sequence {𝑥𝑛 }𝑛≥1 is bounded but does not converge

(D) The sequence {𝑥𝑛 }𝑛≥1 converges

Q.48 Let 𝒪 = {𝑃 ∶ 𝑃 is a 3 × 3 real matrix satisfying 𝑃𝑇 𝑃 = 𝐼3 and det(𝑃) = 1},
where 𝐼3 denotes the identity matrix of order 3. Then which of the following
options is/are correct?

(A) 1
There exists a 𝑃 ∈ 𝒪 with 𝜆 = 2 as an eigen value

(B) There exists a 𝑃 ∈ 𝒪 with 𝜆 = 2 as an eigen value

(C) If 𝜆 is the only real eigen value of 𝑃 ∈ 𝒪, then 𝜆 = 1

(D) There exists a 𝑃 ∈ 𝒪 with 𝜆 = −1 as an eigen value

Page 28 of 36

Q.49 Let 𝑋1 , 𝑋2 , and 𝑋3 be independent standard normal random variables, and let
𝑌1 = 𝑋1 − 𝑋2 , 𝑌2 = 𝑋1 + 𝑋2 − 2𝑋3 and 𝑌3 = 𝑋1 + 𝑋2 + 𝑋3 . Then which of the
following options is/are correct?

(A) 𝑌1 , 𝑌2 and 𝑌3 are independently distributed

(B) 𝑌12 + 𝑌22 + 𝑌32 ∼ 𝜒32

(C) 2𝑌3
∼ 𝑡2
√3𝑌12 + 𝑌22

(D) 3𝑌12 + 2𝑌32
∼ 𝐹1,1
2𝑌22

𝑎.𝑠.
Q.50 Let {𝑋𝑛 }𝑛≥1 be a sequence of independent random variables and 𝑋𝑛 → 0 as 𝑛 → ∞.
Then which of the following options is/are necessarily correct?

(A) E(𝑋𝑛3 ) → 0 as 𝑛 → ∞

(B) 𝑃
𝑋𝑛7 → 0 as 𝑛 → ∞

(C) For any 𝜖 > 0, ∑∞
𝑛=1 Pr(|𝑋𝑛 | ≥ 𝜖) < ∞

𝑎.𝑠.
(D) 𝑋𝑛2 + 𝑋𝑛 + 5 → 5 as 𝑛 → ∞

Page 29 of 36

Q.51 Consider a Markov chain {𝑋𝑛 : 𝑛 = 1, 2, … } with state space 𝑆 = {1, 2, 3} and
transition probability matrix

0 1/2 1/2

𝑃= 1/3 0 2/3 .

(2/5 3/5 0 )
18 24 25
Define 𝜋 = (67 , 67 , 67). Which of the following options is/are correct?

(A) 𝜋 is a stationary distribution of 𝑃

(B) 𝜋 𝑇 is an eigen vector of 𝑃𝑇

(C) 11
Pr(𝑋3 = 1 | 𝑋1 = 1) =
30

(D) At least one state is transient

Q.52 Let 𝑋1 , … , 𝑋𝑛 be a random sample from a uniform distribution over the interval
𝜃 𝜃
(− 2 , 2), where 𝜃 > 0 is an unknown parameter. Then which of the following
options is/are correct?

(A) 2 max{𝑋1 , … , 𝑋𝑛 } is the maximum likelihood estimator of 𝜃

(B) (min{𝑋1 , … , 𝑋𝑛 }, max{𝑋1 , … , 𝑋𝑛 }) is a sufficient statistic

(C) (min{𝑋1 , … , 𝑋𝑛 }, max{𝑋1 , … , 𝑋𝑛 }) is a complete statistic

(D) 2(𝑛+1)
max{|𝑋1 |, … , |𝑋𝑛 |} is a uniformly minimum variance unbiased
𝑛
estimator of 𝜃

Page 30 of 36

Q.53 Let 𝑋 = (𝑋1 , 𝑋2 , 𝑋3 )𝑇 be a 3-dimensional random vector having multivariate
normal distribution with mean vector (0,0,0)𝑇 and covariance matrix

4 0 0
Σ = (0 9 0).
0 0 4
Let 𝛼 𝑇 = (2, 0, −1) and 𝛽 𝑇 = (1, 1, 1). Then which of the following statements
is/are correct?

(A) E(trace(𝑋𝑋 𝑇 𝛼𝛼 𝑇 )) = 20

(B) Var(trace(𝑋𝛼 𝑇 )) = 20

(C) E(trace(𝑋𝑋 𝑇 )) = 17

(D) Cov(𝛼 𝑇 𝑋, 𝛽 𝑇 𝑋) = 3

Page 31 of 36

Q.54 For 𝑌 ∈ ℝ𝑛 , 𝑋 ∈ ℝ𝑛×𝑝 , and 𝛽 ∈ ℝ𝑝 , consider a regression model

𝑌 = 𝑋 𝛽 + 𝜖,

where 𝜖 has an 𝑛-dimensional multivariate normal distribution with zero mean
vector and identity covariance matrix. Let 𝐼𝑝 denote the identity matrix of order 𝑝.
For 𝜆 > 0, let
−1
𝛽̂𝑛 = (𝑋 𝑇 𝑋 + 𝜆𝐼𝑝 ) 𝑋 𝑇 𝑌,

be an estimator of 𝛽. Then which of the following options is/are correct?

(A) 𝛽̂𝑛 is an unbiased estimator of 𝛽

(B) (𝑋 𝑇 𝑋 + 𝜆𝐼𝑝 ) is a positive definite matrix

(C) 𝛽̂𝑛 has a multivariate normal distribution

(D) −1
Var(𝛽̂𝑛 ) = (𝑋 𝑇 𝑋 + 𝜆𝐼𝑝 )

Q.55 Let 𝑓: ℝ2 → ℝ be defined as 𝑓(𝑥, 𝑦) = 𝑥 2 𝑦 2 + 8𝑥 − 4𝑦. The number of saddle
points of 𝑓 is ____________ (answer in integer).

Page 32 of 36

Q.56 Let

0 1 1 1 1
−1 0 1 1 1
𝑃 = −1 −1 0 1 1 .
−1 −1 −1 0 1
(−1 −1 −1 −1 0)

If 𝜆1 , 𝜆2 , 𝜆3 , 𝜆4 , and 𝜆5 are eigen values of 𝑃, then ∏5𝑖=1 𝜆𝑖 equals ____________
(answer in integer).

Q.57 2 1 1 1
Let 𝑃 = ( ) and 𝑄 = ( ). Then the value of trace(𝑃5 + 𝑄 4 ) equals
1 2 −2 4
____________ (answer in integer).

Q.58 The moment generating functions of three independent random variables
𝑋, 𝑌 and 𝑍 are respectively given as

1
𝑀𝑋 (𝑡) = (2 + 𝑒 𝑡 )2 , 𝑡 ∈ ℝ,
9
𝑡
𝑀𝑌 (𝑡) = 𝑒 (𝑒 −1) , 𝑡 ∈ ℝ,
𝑡
and 𝑀𝑍 (𝑡) = 𝑒 2(𝑒 −1) , 𝑡 ∈ ℝ.

Then 10 Pr(𝑋 > 𝑌 + 𝑍) equals __________ (rounded off to two decimal places).

Page 33 of 36

Q.59 The service times (in minutes) at two petrol pumps 𝑃1 and 𝑃2 follow distributions
with probability density functions

𝑓1 (𝑥) = 𝜆𝑒 −𝜆𝑥 , 𝑥 > 0 and 𝑓2 (𝑥) = 𝜆2 𝑥 𝑒 −𝜆𝑥 , 𝑥 > 0,

respectively, where 𝜆 > 0. For service, a customer chooses 𝑃1 or 𝑃2 randomly with
equal probability. Suppose, the probability that the service time for the customer is
more than one minute, is 2𝑒 −2. Then the value of 𝜆 equals ____________ (answer
in integer).

Q.60 Let {𝑋𝑛 }𝑛≥1 be a sequence of independent random variables with

1 1 1
Pr (𝑋𝑛 = − 𝑛
) = Pr (𝑋𝑛 = 𝑛 ) = ∀ 𝑛 ∈ ℕ.
2 2 2
𝑑 2
Suppose that ∑𝑛𝑖=1 𝑋𝑖 → 𝑈 as 𝑛 → ∞. Then 6 Pr (𝑈 ≤ 3) equals ____________
(answer in integer).

Q.61 Let 𝑋1 , 𝑋2 , … , 𝑋7 be a random sample from a population having the probability
density function

1 3 2 −𝜆𝑥
𝑓(𝑥) = 𝜆 𝑥 𝑒 , 𝑥 >0,
2

where 𝜆 > 0 is an unknown parameter. Let 𝜆̂ be the maximum likelihood estimator
of 𝜆, and E(𝜆̂ − 𝜆) = 𝛼𝜆 be the corresponding bias, where 𝛼 is a real constant. Then
1
the value of 𝛼 equals ____________ (answer in integer).

Page 34 of 36

Q.62 Let 𝑋1 , 𝑋2 be a random sample from a population having probability density
function
(𝑥−𝜃)
𝑓𝜃 (𝑥) = {𝑒 if − ∞ < 𝑥 ≤ 𝜃
0 otherwise,

where 𝜃 ∈ ℝ is an unknown parameter. Consider testing 𝐻0 : 𝜃 ≥ 0 against
𝐻1 : 𝜃 < 0 at level 𝛼 = 0.09. Let 𝛽(𝜃) denote the power function of a uniformly
most powerful test. Then 𝛽(log 𝑒 0.36) equals ____________ (rounded off to two
decimal places).

Q.63 Let 𝑋 ∼ 𝐵𝑖𝑛(3, 𝜃), where 𝜃 ∈ (0,1) is an unknown parameter. For testing
1 3 1 3
𝐻0 : 4 ≤ 𝜃 ≤ 4 against 𝐻1 : 𝜃 < 4 or 𝜃 > 4 , consider the test

1 if 𝑥 ∈ {0, 3}
𝜙(𝑥) = {
0 if 𝑥 ∈ {1, 2}.

The size of the test 𝜙 is __________ (rounded off to two decimal places).

Q.64 Let (𝑋1 , 𝑋2 , 𝑋3 )𝑇 have the following distribution

0 1 0.4 0
𝑁3 ((0), (0.4 1 0.6) ).
0 0 0.6 1

Then the value of the partial correlation coefficient between 𝑋1 and 𝑋2 given 𝑋3 is
__________ (rounded off to two decimal places).

Page 35 of 36

Q.65 Let (𝑋, 𝑌)𝑇 follow a bivariate normal distribution with E(𝑋) = 2, E(𝑌) = 3,
Var(𝑋) = 16, Var(𝑌) = 25, and Cov(𝑋, 𝑌) = 14. Then

1
2𝜋 (Pr(𝑋 > 2, 𝑌 > 3) − )
4

equals ____________ (rounded off to two decimal places).

Page 36 of 36

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