GATE 2021 Question Paper ST Statistics

General Aptitude (GA)

Q.1 – Q.5 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong
answer: – 1/3).

Q.1 The current population of a city is 11,02,500. If it has been increasing at the
rate of 5% per annum, what was its population 2 years ago?

(A) 9,92,500

(B) 9,95,006

(C) 10,00,000

(D) 12,51,506

Q.2 p q
p and q are positive integers and + = 3,
q p

p2 q2
then, 2
+ =
q p2

(A) 3

(B) 7

(C) 9

(D) 11

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Q.3

The least number of squares that must be added so that the line P-Q
becomes the line of symmetry is ________

(A) 4

(B) 3

(C) 6

(D) 7

Q.4 Nostalgia is to anticipation as _______ is to ________
Which one of the following options maintains a similar logical relation in the
above sentence?

(A) Present, past

(B) Future, past

(C) Past, future

(D) Future, present

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Q.5 Consider the following sentences:
(i) I woke up from sleep.
(ii) I woked up from sleep.
(iii) I was woken up from sleep.
(iv) I was wokened up from sleep.
Which of the above sentences are grammatically CORRECT?

(A) (i) and (ii)

(B) (i) and (iii)

(C) (ii) and (iii)

(D) (i) and (iv)

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Q. 6 – Q. 10 Multiple Choice Question (MCQ), carry TWO marks each (for each wrong
answer: – 2/3).

Q.6 Given below are two statements and two conclusions.
Statement 1: All purple are green.
Statement 2: All black are green.
Conclusion I: Some black are purple.
Conclusion II: No black is purple.
Based on the above statements and conclusions, which one of the following
options is logically CORRECT?

(A) Only conclusion I is correct.

(B) Only conclusion II is correct.

(C) Either conclusion I or II is correct.

(D) Both conclusion I and II are correct.

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Q.7 Computers are ubiquitous. They are used to improve efficiency in almost all
fields from agriculture to space exploration. Artificial intelligence (AI) is
currently a hot topic. AI enables computers to learn, given enough training
data. For humans, sitting in front of a computer for long hours can lead to
health issues.
Which of the following can be deduced from the above passage?
(i) Nowadays, computers are present in almost all places.
(ii) Computers cannot be used for solving problems in engineering.
(iii) For humans, there are both positive and negative effects of using
computers.
(iv) Artificial intelligence can be done without data.

(A) (ii) and (iii)

(B) (ii) and (iv)

(C) (i), (iii) and (iv)

(D) (i) and (iii)

Q.8 Consider a square sheet of side 1 unit. In the first step, it is cut along the
main diagonal to get two triangles. In the next step, one of the cut triangles
is revolved about its short edge to form a solid cone. The volume of the
resulting cone, in cubic units, is ________

(A) 𝜋
3
(B) 2𝜋
3
(C) 3𝜋
2
(D) 3π

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Q.9
65
Sunday 55

50
Saturday 60

35
Friday 20

55
Thursday 60

50
Wednesday 60

65
Tuesday 55

70
Monday 45

0 10 20 30 40 50 60 70 80
Y X

The number of minutes spent by two students, X and Y, exercising every day
in a given week are shown in the bar chart above.
The number of days in the given week in which one of the students spent a
minimum of 10% more than the other student, on a given day, is

(A) 4

(B) 5

(C) 6

(D) 7

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Q.10

Corners are cut from an equilateral triangle to produce a regular convex
hexagon as shown in the figure above.
The ratio of the area of the regular convex hexagon to the area of the original
equilateral triangle is

(A) 2:3

(B) 3:4

(C) 4:5

(D) 5:6

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Q.1 – Q.9 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong
answer: – 1/3).

Q.1 Let 𝑿 be a non-constant positive random variable such that 𝑬(𝑿) = 𝟗.
Then which one of the following statements is true?

(A) 𝐸 ( 1 ) > 0.1 and 𝑃(𝑋 ≥ 10) ≤ 0.9
𝑋+1

(B) 𝐸 ( 1 ) < 0.1 and 𝑃(𝑋 ≥ 10) ≤ 0.9
𝑋+1

(C ) 𝐸 ( 1 ) > 0.1 and 𝑃(𝑋 ≥ 10) > 0.9
𝑋+1

(D) 𝐸 ( 1 ) < 0.1 and 𝑃(𝑋 ≥ 10) > 0.9
𝑋+1

Q.2 Let {𝑾(𝒕)}𝒕≥𝟎 be a standard Brownian motion. Then the variance of
𝑾(𝟏)𝑾(𝟐) equals

(A) 1

(B) 2

(C) 3

(D) 4

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Q.3 Let 𝑿𝟏 , 𝑿𝟐 , … , 𝑿𝒏 be a random sample of size 𝒏 (≥ 𝟐) from a distribution
having the probability density function
1 − 𝑥−𝜃
𝑓(𝑥; 𝜃) = { 𝜃 𝑒
𝜃 , 𝑥 > 𝜃,
0, otherwise,
where 𝜃 ∈ (0, ∞). Then the method of moments estimator of 𝜃 equals
𝑛
(A) 1
∑ 𝑋𝑖
2𝑛
𝑖=1

𝑛
(B) 2
∑ 𝑋𝑖
𝑛
𝑖=1

𝑛
(C) 1
∑ 𝑋𝑖
𝑛
𝑖=1

(D) 𝑛
∑𝑛𝑖=1 𝑋𝑖

Q.4 Let {𝒙𝟏 , 𝒙𝟐 , … , 𝒙𝒏 } be a realization of a random sample of size 𝒏 (≥ 𝟐)
from a 𝑵(𝝁, 𝝈𝟐 ) distribution, where −∞ < 𝝁 < ∞ and 𝝈 > 𝟎. Which of
the following statements is/are true?
P : 95% confidence interval of 𝜇 based on {𝑥1 , 𝑥2 , … , 𝑥𝑛 } is unique when
𝜎 is known.
Q : 95% confidence interval of 𝜇 based on {𝑥1 , 𝑥2 , … , 𝑥𝑛 } is NOT unique
when 𝜎 is unknown.

(A) P only

(B) Q only

(C) Both P and Q

(D) Neither P nor Q

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Q.5 Let 𝑿𝟏 , 𝑿𝟐 , … , 𝑿𝒏 be a random sample of size 𝒏 (≥ 𝟐) from a 𝑵(𝟎, 𝝈𝟐 )
distribution. For a given 𝝈 > 𝟎, let 𝒇𝝈 denote the joint probability density
function of (𝑿𝟏 , 𝑿𝟐 , … , 𝑿𝒏 ) and 𝑺 = {𝒇𝝈 : 𝝈 > 𝟎}. Let 𝑻𝟏 = ∑𝒏𝒊=𝟏 𝑿𝟐𝒊 and
𝟏 𝟐
𝑻𝟐 = (𝒏 ∑𝒏𝒊=𝟏 𝑿𝒊 ) . For any positive integer 𝝂 and any 𝜶 ∈ (𝟎, 𝟏), let 𝝌𝟐𝝂,𝜶
denote the (𝟏 − 𝜶)-th quantile of the central chi-square distribution with 𝝂
degrees of freedom. Consider testing 𝑯𝟎 : 𝝈 = 𝟏 against 𝑯𝟏 : 𝝈 > 𝟏 at level
𝜶. Then which one of the following statements is true?
2
(A) 𝑆 has a monotone likelihood ratio in 𝑇1 and 𝐻0 is rejected if 𝑇1 > 𝜒𝑛,𝛼
2
(B) 𝑆 has a monotone likelihood ratio in 𝑇1 and 𝐻0 is rejected if 𝑇1 > 𝜒𝑛,1−𝛼
2
(C) 𝑆 has a monotone likelihood ratio in 𝑇2 and 𝐻0 is rejected if 𝑇2 > 𝜒𝑛,𝛼
2
(D) 𝑆 has a monotone likelihood ratio in 𝑇2 and 𝐻0 is rejected if 𝑇2 > 𝜒𝑛,1−𝛼

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Q.6 Let 𝑿 and 𝒀 be two random variables such that 𝒑𝟏𝟏 + 𝒑𝟏𝟎 + 𝒑𝟎𝟏 + 𝒑𝟎𝟎 =
𝟏, where 𝒑𝒊𝒋 = 𝑷(𝑿 = 𝒊, 𝒀 = 𝒋), 𝒊, 𝒋 = 𝟎, 𝟏. Suppose that a realization of a
random sample of size 𝟔𝟎 from the joint distribution of (𝑿, 𝒀) gives
𝒏𝟏𝟏 = 𝟏𝟎, 𝒏𝟏𝟎 = 𝟐𝟎, 𝒏𝟎𝟏 = 𝟐𝟎 𝐚𝐧𝐝 𝒏𝟎𝟎 = 𝟏𝟎,
where 𝒏𝒊𝒋 denotes the frequency of (𝒊, 𝒋) for 𝒊, 𝒋 = 𝟎, 𝟏. If the chi-square
test of independence is used to test
𝑯𝟎 : 𝒑𝒊𝒋 = 𝒑𝒊⋅ 𝒑⋅𝒋 for 𝒊, 𝒋 = 𝟎, 𝟏 against 𝑯𝟏 : 𝒑𝒊𝒋 ≠ 𝒑𝒊⋅ 𝒑⋅𝒋 for at least one pair
(𝒊, 𝒋),
where 𝒑𝒊⋅ = 𝒑𝒊𝟎 + 𝒑𝒊𝟏 and 𝒑⋅𝒋 = 𝒑𝟎𝒋 + 𝒑𝟏𝒋 , then which one of the following
statements is true?

(A) Under 𝐻0 , the test statistic follows central chi-square distribution with one
20
degree of freedom and the observed value of the test statistic is 3

(B) Under 𝐻0 , the test statistic follows central chi-square distribution with three
20
degrees of freedom and the observed value of the test statistic is 3

(C) Under 𝐻0 , the test statistic follows central chi-square distribution with one
16
degree of freedom and the observed value of the test statistic is 3

(D) Under 𝐻0 , the test statistic follows central chi-square distribution with three
16
degrees of freedom and the observed value of the test statistic is 3

Q.7 Let the joint distribution of (𝑿, 𝒀) be bivariate normal with mean vector
𝟎 𝟏 𝝆
( ) and variance-covariance matrix ( ), where −𝟏 < 𝝆 < 𝟏. Let
𝟎 𝝆 𝟏
𝚽𝝆 (𝟎, 𝟎) = 𝑷(𝑿 ≤ 𝟎, 𝒀 ≤ 𝟎). Then the Kendall’s 𝝉 coefficient between 𝑿
and 𝒀 equals

(A) 4Φ𝜌 (0,0) − 1

(B) 4Φ𝜌 (0,0)

(C) 4Φ𝜌 (0,0) + 1

(D) Φ𝜌 (0,0)

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Q.8 Consider the simple linear regression model
𝒀𝒊 = 𝜷𝟎 + 𝜷𝟏 𝒙𝒊 + 𝝐𝒊 , 𝒊 = 𝟏, 𝟐, … , 𝒏 (𝒏 ≥ 𝟑),
where 𝜷𝟎 and 𝜷𝟏 are unknown parameters and 𝝐𝒊 ’s are independent and
identically distributed random variables with mean zero and finite variance
𝝈𝟐 > 𝟎. Suppose that 𝜷 ̂𝟎 and 𝜷̂𝟏 are the ordinary least squares estimators
𝟏
of 𝜷𝟎 and 𝜷𝟏 , respectively. Define 𝒙̅ = ∑𝒏𝒊=𝟏 𝒙𝒊 , 𝑺𝟏 = ∑𝒏𝒊=𝟏(𝒙𝒊 − 𝒙
̅)𝟐 and
𝒏
𝑺𝟐 = ∑𝒏𝒊=𝟏 𝒚𝒊 (𝒙𝒊 − 𝒙
̅), where 𝒚𝒊 is the observed value of 𝒀𝒊 , 𝒊 = 𝟏, 𝟐, … , 𝒏.
̂𝟎 + 𝒄 is
Then for a real constant 𝒄, the variance of 𝜷

(A) 2
1 𝑥̅ 2
𝜎 ( + )
𝑛 𝑆2

(B) 2
1 𝑥̅ 2
𝜎 ( + )
𝑛 𝑆1

(C) 𝜎 2
𝑛
(D) 1 𝑥̅ 2
𝜎2 ( + ) + 𝑐2
𝑛 𝑆2

Q.9 Let 𝑿𝟏 , 𝑿𝟐 , 𝑿𝟑 , 𝒀𝟏 , 𝒀𝟐 , 𝒀𝟑 and 𝒀𝟒 be independent random vectors such that
𝑿𝒊 follows 𝑵𝟒 (𝟎, 𝚺𝟏 ) distribution for 𝒊 = 𝟏, 𝟐, 𝟑, and 𝒀𝒋 follows 𝑵𝟒 (𝟎, 𝚺𝟐 )
distribution for 𝒋 = 𝟏, 𝟐, 𝟑, 𝟒, where 𝚺𝟏 and 𝚺𝟐 are positive definite
matrices. Further, let
𝒁 = 𝚺𝟏− ½ 𝑿𝑿𝑻 𝚺𝟏− ½ + 𝚺𝟐− ½ 𝒀𝒀𝑻 𝚺𝟐− ½ ,
where 𝑿 = [𝑿𝟏 𝑿𝟐 𝑿𝟑 ] is a 𝟒 × 𝟑 matrix, 𝒀 = [𝒀𝟏 𝒀𝟐 𝒀𝟑 𝒀𝟒 ] is a 𝟒 × 𝟒
matrix and 𝑿𝑻 and 𝒀𝑻 denote transposes of 𝑿 and 𝒀, respectively. If
𝑾𝒎 (𝒏, 𝚺) denotes a Wishart distribution of order 𝒎 with 𝒏 degrees of
freedom and variance-covariance matrix 𝚺 and 𝑰𝒏 denotes the 𝒏 × 𝒏
identity matrix, then which one of the following statements is true?

(A) 𝑍 follows 𝑊4 (7, 𝐼4 ) distribution

(B) 𝑍 follows 𝑊4 (4, 𝐼4 ) distribution

(C) 𝑍 follows 𝑊7 (4, 𝐼7 ) distribution

(D) 𝑍 follows 𝑊7 (7, 𝐼7 ) distribution

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Q.10 – Q.25 Numerical Answer Type (NAT), carry ONE mark each (no negative
marks).

Q.10 𝟏
𝒏 𝟏
𝐥𝐢𝐦 (𝟐𝒏 + 𝒏𝟐𝒏 𝐬𝐢𝐧𝟐 ) 𝟐𝒏−𝒏 𝐜𝐨𝐬𝒏
𝒏→∞ 𝟐
equals __________

Q.11 Let
𝟏 𝒙
𝟏
𝑰=𝟒 ∫ ∫ 𝒅𝒚 𝒅𝒙
𝟏 √𝟏−𝒙𝟐
√𝒙𝟐 + 𝒚𝟐
√𝟐
Then the value of 𝒆𝑰+𝝅 equals __________ (round off to 𝟐 decimal
places).

Q.12 𝟎 𝟎 𝟏
Let 𝑨 = [𝟏 𝟎 𝟎] and 𝑰𝟑 be the 𝟑 × 𝟑 identity matrix. Then the
𝟎 𝟏 𝟎
nullity of 𝟓𝑨(𝑰𝟑 + 𝑨 + 𝑨𝟐 ) equals __________

Q.13 Let 𝑨 be the 𝟐 × 𝟐 real matrix having eigenvalues 𝟏 𝐚𝐧𝐝 − 𝟏, with
√𝟑 −𝟏
𝟐 𝟐
corresponding eigenvectors [ ] and [ ], respectively. If 𝑨𝟐𝟎𝟐𝟏 =
𝟏 √𝟑
𝟐 𝟐
𝒂 𝒃
[ ], then 𝒂 + 𝒃 + 𝒄 + 𝒅 equals __________ (round off to 𝟐 decimal
𝒄 𝒅
places).

Q.14 𝟑 𝟏
Let 𝑨 and 𝑩 be two events such that 𝑷(𝑩) = and 𝑷(𝑨 ∪ 𝑩𝒄 ) = .
𝟒 𝟐
If 𝑨 and 𝑩 are independent, then 𝑷(𝑨) equals __________ (round off to
𝟐 decimal places).

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Q.15 A fair die is rolled twice independently. Let 𝑿 and 𝒀 denote the outcomes
of the first and second roll, respectively. Then 𝑬(𝑿 + 𝒀 | (𝑿 − 𝒀)𝟐 = 𝟏)
equals __________

Q.16 Let 𝑿 be a random variable having distribution function
𝟎, 𝒙 < 𝟏,
𝒂
, 𝟏 ≤ 𝒙 < 𝟐,
𝑭(𝒙) = 𝟐𝒄
, 𝟐 ≤ 𝒙 < 𝟑,
𝟔
{ 𝟏, 𝒙 ≥ 𝟑,
𝟏 𝟏
where 𝒂 and 𝒄 are appropriate constants. Let 𝑨𝒏 = [𝟏 + 𝒏 , 𝟑 − 𝒏], 𝒏 ≥
𝟏 𝟓
𝟏, and 𝑨 = ⋃∞
𝒊=𝟏 𝑨𝒊 . If 𝑷(𝑿 ≤ 𝟏) = 𝟐 and 𝑬(𝑿) = 𝟑, then 𝑷(𝑿 ∈ 𝑨)
equals __________ (round off to 𝟐 decimal places).

Q.17 If the marginal probability density function of the 𝒌𝒕𝒉 order statistic of a
random sample of size 8 from a uniform distribution on [𝟎, 𝟐] is
𝟕 𝟔
(𝟐 − 𝒙), 𝟎 < 𝒙 < 𝟐,
𝒇(𝒙) = { 𝟑𝟐 𝒙
𝟎, 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞,
then 𝒌 equals __________

Q.18 (𝜶)
For 𝜶 > 𝟎, let {𝑿𝒏 } be a sequence of independent random variables
𝒏≥𝟏
such that
(𝜶) 𝟏 (𝜶)
𝑷(𝑿𝒏 = 𝟏) = = 𝟏 − 𝑷(𝑿𝒏 = 𝟎).
𝒏𝟐𝜶
(𝜶)
Let 𝑺 = {𝜶 > 𝟎 ∶ 𝑿𝒏 𝐜𝐨𝐧𝐯𝐞𝐫𝐠𝐞𝐬 𝐭𝐨 𝟎 𝐚𝐥𝐦𝐨𝐬𝐭 𝐬𝐮𝐫𝐞𝐥𝐲 𝐚𝐬 𝒏 → ∞}. Then
the infimum of 𝑺 equals __________ (round off to 𝟐 decimal places).

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Q.19 Let {𝑿𝒏 }𝒏≥𝟏 be a sequence of independent and identically distributed
random variables each having uniform distribution on [𝟎, 𝟐]. For 𝒏 ≥ 𝟏,
let
𝟏
𝒏 𝒏
𝒁𝒏 = − 𝐥𝐨𝐠 𝒆 (∏(𝟐 − 𝑿𝒊 )) .
𝒊=𝟏

Then, as 𝒏 → ∞, the sequence {𝒁𝒏 }𝒏≥𝟏 converges almost surely to
_________ (round off to 𝟐 decimal places).

Q.20 Let {𝑿𝒏 }𝒏≥𝟎 be a time-homogeneous discrete time Markov chain with state
space {𝟎, 𝟏} and transition probability matrix
𝟎. 𝟐𝟓 𝟎. 𝟕𝟓
[ ].
𝟎. 𝟕𝟓 𝟎. 𝟐𝟓
If 𝑷(𝑿𝟎 = 𝟎) = 𝑷(𝑿𝟎 = 𝟏) = 𝟎. 𝟓, then
𝟏𝟎𝟎

∑ 𝑬[(𝑿𝟐𝒌 )𝟐𝒌 ]
𝒌=𝟏
equals __________

Q.21 Let {𝟎, 𝟐} be a realization of a random sample of size 𝟐 from a binomial
𝟏
distribution with parameters 𝟐 𝐚𝐧𝐝 𝒑, where 𝒑 ∈ (𝟎, 𝟏). To test 𝑯𝟎 : 𝒑 = 𝟐
𝟏
against 𝑯𝟏 : 𝒑 ≠ 𝟐, the observed value of the likelihood ratio test statistic
equals __________ (round off to 𝟐 decimal places).

Q.22 Let 𝑿 be a random variable having the probability density function
𝟑
(𝟏
𝒇(𝒙) = { 𝟏𝟑 − 𝒙)(𝟗 − 𝒙), 𝟎 < 𝒙 < 𝟏,
𝟎, 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞.
𝟒
Then 𝑬[𝑿(𝑿𝟐 − 𝟏𝟓𝑿 + 𝟐𝟕)] equals __________ (round off to 𝟐
𝟑
decimal places).

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Q.23 𝟓
Let (𝒀, 𝑿𝟏 , 𝑿𝟐 ) be a random vector with mean vector (𝟐) and variance-
𝟎
𝟏𝟎 𝟎. 𝟓 −𝟎. 𝟓
covariance matrix [ 𝟎. 𝟓 𝟕 𝟏. 𝟓 ]. Then the value of the multiple
−𝟎. 𝟓 𝟏. 𝟓 𝟐
correlation coefficient between 𝒀 and its best linear predictor on 𝑿𝟏 and
𝑿𝟐 equals __________ (round off to 𝟐 decimal places).

Q.24 Let 𝑿𝟏 , 𝑿𝟐 and 𝑿𝟑 be a random sample from a bivariate normal
distribution with unknown mean vector 𝝁 and unknown variance-
covariance matrix 𝚺, which is a positive definite matrix. The 𝒑-value
corresponding to the likelihood ratio test for testing 𝑯𝟎 : 𝝁 = 𝟎 against
𝟏 𝟒 −𝟓
𝑯𝟏 : 𝝁 ≠ 𝟎 based on the realization {( ) , ( ) , ( )} of the random
𝟐 −𝟐 𝟎
sample equals __________ (round off to 𝟐 decimal places).

Q.25 Let 𝒀𝒊 = 𝜶 + 𝜷𝒙𝒊 + 𝝐𝒊 , 𝒊 = 𝟏, 𝟐, 𝟑, where 𝒙𝒊 ’s are fixed covariates, 𝜶 and
𝜷 are unknown parameters and 𝝐𝒊 ’s are independent and identically
distributed random variables with mean zero and finite variance. Let 𝜶 ̂ and
̂ be the ordinary least squares estimators of 𝜶 and 𝜷, respectively. Given
𝜷
the following observations

𝒚𝒊 𝟖. 𝟔𝟐 𝟐𝟔. 𝟖𝟔 𝟓𝟒. 𝟎𝟐

𝒙𝒊 𝟑. 𝟐𝟗 𝟐𝟏. 𝟓𝟑 𝟒𝟖. 𝟔𝟗

̂ equals __________ (round off to 𝟐 decimal places).
̂+𝜷
the value of 𝜶

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Q.26 – Q.43 Multiple Choice Question (MCQ), carry TWO mark each (for each wrong
answer: – 2/3).

Q.26 Let 𝒇: ℝ → ℝ be defined by
𝒙𝟑 𝐬𝐢𝐧 𝒙, 𝒙 = 𝟎 𝐨𝐫 𝒙 𝐢𝐬 𝐢𝐫𝐫𝐚𝐭𝐢𝐨𝐧𝐚𝐥,
𝒇(𝒙) = { 𝟏 𝒑
𝟑
, 𝒙 = , 𝒑 ∈ ℤ ∖ {𝟎}, 𝒒 ∈ ℕ 𝐚𝐧𝐝 𝐠𝐜𝐝(𝒑, 𝒒) = 𝟏,
𝒒 𝒒
where ℝ denotes the set of all real numbers, ℤ denotes the set of all integers,
ℕ denotes the set of all positive integers and 𝐠𝐜𝐝(𝒑, 𝒒) denotes the greatest
common divisor of 𝒑 and 𝒒. Then which one of the following statements is
true?

(A) 𝑓 is not continuous at 0

(B) 𝑓 is not differentiable at 0

(C) 𝑓 is differentiable at 0 and the derivative of 𝑓 at 0 equals 0

(D) 𝑓 is differentiable at 0 and the derivative of 𝑓 at 0 equals 1

Q.27 Let 𝒇: [𝟎, ∞) → ℝ be a function, where ℝ denotes the set of all real
numbers. Then which one of the following statements is true?

(A) If 𝑓 is bounded and continuous, then 𝑓 is uniformly continuous

(B) If 𝑓 is uniformly continuous, then lim 𝑓(𝑥) exists
𝑥→∞

(C) If 𝑓 is uniformly continuous, then the function 𝑔(𝑥) = 𝑓(𝑥) sin 𝑥 is also
uniformly continuous

(D) If 𝑓 is continuous and lim 𝑓(𝑥) is finite, then 𝑓 is uniformly continuous
𝑥→∞

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Q.28 Let 𝒇: ℝ → ℝ be a differentiable function such that 𝒇(𝟎) = 𝟎 and
𝒇′ (𝒙) + 𝟐𝒇(𝒙) > 𝟎 for all 𝒙 ∈ ℝ, where 𝒇′ denotes the derivative of 𝒇 and
ℝ denotes the set of all real numbers. Then which one of the following
statements is true?

(A) 𝑓(𝑥) > 0, for all 𝑥 > 0 and 𝑓(𝑥) < 0, for all 𝑥 < 0

(B) 𝑓(𝑥) < 0, for all 𝑥 ≠ 0

(C) 𝑓(𝑥) > 0, for all 𝑥 ≠ 0

(D) 𝑓(𝑥) < 0, for all 𝑥 > 0 and 𝑓(𝑥) > 0, for all 𝑥 < 0

Q.29 Let 𝑴 be the collection of all 𝟑 × 𝟑 real symmetric positive definite
matrices. Consider the set
𝟏
𝑺 = {𝑨 ∈ 𝑴 ∶ 𝑨𝟓𝟎 − 𝑨𝟒𝟖 = 𝟎},
𝟒
where 𝟎 denotes the 𝟑 × 𝟑 zero matrix. Then the number of elements in
𝑺 equals

(A) 0

(B) 1

(C) 8

(D) ∞

Q.30 Let 𝑨 be a 𝟑 × 𝟑 real matrix such that 𝑰𝟑 + 𝑨 is invertible and let
𝑩 = (𝑰𝟑 + 𝑨)−𝟏 (𝑰𝟑 − 𝑨), where 𝑰𝟑 denotes the 𝟑 × 𝟑 identity matrix. Then
which one of the following statements is true?

(A) If 𝐵 is orthogonal, then 𝐴 is invertible

(B) If 𝐵 is orthogonal, then all the eigenvalues of 𝐴 are real

(C) If 𝐵 is skew-symmetric, then 𝐴 is orthogonal

(D) If 𝐵 is skew-symmetric, then the determinant of 𝐴 equals −1

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Q.31 Let 𝑿 be a random variable having Poisson distribution such that
𝑬(𝑿𝟐 ) = 𝟏𝟏𝟎. Then which one of the following statements is NOT true?

(A) 𝐸(𝑋 𝑛 ) = 10 𝐸[(𝑋 + 1)𝑛−1 ], for all 𝑛 = 1, 2, 3, …

(B) 1
𝑃(𝑋 is even) = 4 (1 + 𝑒 −20 )

(C) 𝑃(𝑋 = 𝑘) < 𝑃(𝑋 = 𝑘 + 1), for 𝑘 = 0, 1, … , 8

(D) 𝑃(𝑋 = 𝑘) > 𝑃(𝑋 = 𝑘 + 1), for 𝑘 = 10, 11, …

Q.32 𝝅 𝝅
Let 𝑿 be a random variable having uniform distribution on [− 𝟐 , 𝟐 ].
Then which one of the following statements is NOT true?

(A) 𝑌 = cot 𝑋 follows standard Cauchy distribution

(B) 𝑌 = tan 𝑋 follows standard Cauchy distribution

(C) 𝑌 = − log ( 1 + 𝑋) has moment generating function 𝑀 (𝑡 ) = 1
𝑒 2 𝜋
, 𝑡<1
1−𝑡

(D) 𝑌 = −2 log ( 1 + 𝑋) follows central chi-square distribution with one degree of
𝑒 2 𝜋
freedom

Q.33 Let 𝛀 = {𝟏, 𝟐, 𝟑, … } represent the collection of all possible outcomes of a
random experiment with probabilities 𝑷({𝒏}) = 𝜶𝒏 for 𝒏 ∈ 𝛀. Then which
one of the following statements is NOT true?

(A) lim 𝛼𝑛 = 0
𝑛→∞

(B) ∑∞
𝑛=1 √𝛼𝑛 converges

(C) For any positive integer 𝑘, there exist 𝑘 disjoint events 𝐴1 , 𝐴2 , … , 𝐴𝑘 such
that 𝑃(⋃𝑘𝑖=1 𝐴𝑖 ) < 0.001

(D) There exists a sequence {𝐴𝑖 }𝑖≥1 of strictly increasing events such that
𝑃(⋃∞𝑖=1 𝐴𝑖 ) < 0.001

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Q.34 Let (𝑿, 𝒀) have the joint probability density function
𝟒
, 𝒙 > 𝟏, 𝒚 > 𝟏,
𝒇𝑿,𝒀 (𝒙, 𝒚) = {(𝒙 + 𝒚)𝟑
𝟎, 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞.
Then which one of the following statements is NOT true?

(A) The probability density function of 𝑋 + 𝑌 is
4
(𝑧
𝑓𝑋+𝑌 (𝑧) = { 𝑧 3 − 2), 𝑧 > 2,
0, otherwise.

(B) 3
𝑃(𝑋 + 𝑌 > 4) =
4
(C) 𝐸(𝑋 + 𝑌) = 4 log 𝑒 2

(D) 𝐸(𝑌 | 𝑋 = 2) = 4

Q.35 Let 𝑿𝟏 , 𝑿𝟐 and 𝑿𝟑 be three uncorrelated random variables with common
variance 𝝈𝟐 < ∞. Let 𝒀𝟏 = 𝟐𝑿𝟏 + 𝑿𝟐 + 𝑿𝟑 , 𝒀𝟐 = 𝑿𝟏 + 𝟐𝑿𝟐 + 𝑿𝟑 and
𝒀𝟑 = 𝑿𝟏 + 𝑿𝟐 + 𝟐𝑿𝟑 . Then which of the following statements is/are true?
P : The sum of eigenvalues of the variance covariance matrix of
(𝒀𝟏 , 𝒀𝟐 , 𝒀𝟑 )
is 𝟏𝟖𝝈𝟐 .
Q : The correlation coefficient between 𝒀𝟏 𝐚𝐧𝐝 𝒀𝟐 equals that between
𝒀𝟐 𝐚𝐧𝐝 𝒀𝟑 .

(A) P only

(B) Q only

(C) Both P and Q

(D) Neither P nor Q

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Q.36 Let {𝑿𝒏 }𝒏≥𝟎 be a time-homogeneous discrete time Markov chain with either
finite or countable state space 𝑺. Then which one of the following statements
is true?

(A) There is at least one recurrent state

(B) If there is an absorbing state, then there exists at least one stationary distribution

(C) If all the states are positive recurrent, then there exists a unique stationary
distribution

(D) If {𝑋𝑛 }𝑛≥0 is irreducible, 𝑆 = {1, 2} and [𝜋1 𝜋2 ] is a stationary distribution,
then lim 𝑃(𝑋𝑛 = 𝑖 | 𝑋0 = 𝑖) = 𝜋𝑖 for 𝑖 = 1, 2
𝑛→∞

Q.37 Let customers arrive at a departmental store according to a Poisson process
with rate 𝟏𝟎. Further, suppose that each arriving customer is either a male
𝟏
or a female with probability 𝟐 each, independent of all other arrivals. Let
𝑵(𝒕) denote the total number of customers who have arrived by time 𝒕.
Then which one of the following statements is NOT true?

(A) If 𝑆2 denotes the time of arrival of the second female customer, then
1
𝑃(𝑆2 ≤ 1) = 25 ∫0 𝑠𝑒 −5𝑠 𝑑𝑠

(B) If 𝑀(𝑡) denotes the number of male customers who have arrived by time 𝑡,
1 1
then 𝑃 (𝑀 (3) = 0 | 𝑀(1) = 1) = 3

(C) 𝐸 [(𝑁(𝑡))2 ] = 100𝑡 2 + 10𝑡

(D) 𝐸[𝑁(𝑡)𝑁(2𝑡)] = 200𝑡 2 + 10𝑡

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Q.38 Let 𝑿(𝟏) < 𝑿(𝟐) < 𝑿(𝟑) < 𝑿(𝟒) < 𝑿(𝟓) be the order statistics corresponding
to a random sample of size 𝟓 from a uniform distribution on [𝟎, 𝜽], where
𝜽 ∈ (𝟎, ∞). Then which of the following statements is/are true?
P : 𝟑𝑿(𝟐) is an unbiased estimator of 𝜽.
Q : The variance of 𝑬[𝟐𝑿(𝟑) | 𝑿(𝟓) ] is less than or equal to the variance
of 𝟐𝑿(𝟑) .

(A) P only

(B) Q only

(C) Both P and Q

(D) Neither P nor Q

Q.39 Let 𝑿𝟏 , 𝑿𝟐 , … , 𝑿𝒏 be a random sample of size 𝒏 (≥ 𝟐) from a
distribution having the probability density function
𝟏 −𝒙
𝒇(𝒙; 𝜽) = {𝜽 𝒆 , 𝒙 > 𝟎,
𝜽

𝟎, 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞,
where 𝜽 ∈ (𝟎, ∞). Let 𝑿(𝟏) = 𝐦𝐢𝐧{ 𝑿𝟏 , 𝑿𝟐 , … , 𝑿𝒏 } and 𝑻 = ∑𝒏𝒊=𝟏 𝑿𝒊 .
Then 𝑬(𝑿(𝟏) | 𝑻 ) equals

(A) 𝑇
𝑛2
(B) 𝑇
𝑛
(C) (𝑛 + 1)𝑇
2𝑛
(D) (𝑛 + 1)2 𝑇
4𝑛2

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Q.40 Let 𝑿𝟏 , 𝑿𝟐 , … , 𝑿𝒏 be a random sample of size 𝒏 (≥ 𝟐) from a uniform
distribution on [−𝜽, 𝜽], where 𝜽 ∈ (𝟎, ∞). Let 𝑿(𝟏) = 𝐦𝐢𝐧{ 𝑿𝟏 , 𝑿𝟐 , … , 𝑿𝒏 }
and 𝑿(𝒏) = 𝐦𝐚𝐱{ 𝑿𝟏 , 𝑿𝟐 , … , 𝑿𝒏 }. Then which of the following statements
is/are true?
P : (𝑿(𝟏), 𝑿(𝒏) ) is a complete statistic.
Q : 𝑿(𝒏) − 𝑿(𝟏) is an ancillary statistic.

(A) P only

(B) Q only

(C) Both P and Q

(D) Neither P nor Q

Q.41 Let {𝑿𝒏 }𝒏≥𝟏 be a sequence of independent and identically distributed
random variables having common distribution function 𝑭(⋅). Let 𝒂 < 𝒃 be
two real numbers such that 𝑭(𝒙) = 𝟎 for all 𝒙 ≤ 𝒂, 𝟎 < 𝑭(𝒙) < 𝟏 for all
𝒂 < 𝒙 < 𝒃 and 𝑭(𝒙) = 𝟏 for all 𝒙 ≥ 𝒃. Let 𝑺𝒏 (𝒙) be the empirical
distribution function at 𝒙 based on 𝑿𝟏 , 𝑿𝟐 , … , 𝑿𝒏 , 𝒏 ≥ 𝟏. Then which one
of the following statements is NOT true?

(A) 𝑃 [ lim sup |𝑆𝑛 (𝑥) − 𝐹(𝑥)| = 0] = 1
𝑛→∞ −∞<𝑥<∞

(B) For fixed 𝑥 ∈ (𝑎, 𝑏) and 𝑡 ∈ (− ∞, ∞),

√𝑛 |𝑆𝑛 (𝑥) − 𝐹(𝑥)|
lim 𝑃 ≤ 𝑡 = 𝑃(𝑍 ≤ 𝑡),
𝑛→∞
√𝑆 (𝑥)(1 − 𝑆𝑛 (𝑥))
[ 𝑛 ]
where 𝑍 is the standard normal random variable

(C) The covariance between 𝑆𝑛 (𝑥) and 𝑆𝑛 (𝑦) equals 1 𝐹(𝑥)(1 − 𝐹(𝑦)) for all
𝑛
𝑛 ≥ 2 and for fixed −∞ < 𝑥, 𝑦 < ∞

(D) If 𝑌 = 2
𝑛 sup (𝑆𝑛 (𝑥) − 𝐹(𝑥)) , then {4𝑛 𝑌𝑛 }𝑛≥1 converges in distribution to
−∞<𝑥<∞
a central chi-square random variable with 2 degrees of freedom

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Q.42 Let the joint distribution of random variables 𝑿𝟏 , 𝑿𝟐 , 𝑿𝟑 and 𝑿𝟒 be
𝑵𝟒 ( 𝝁, 𝚺), where
𝟏 𝟏 𝟎. 𝟐 𝟎 𝟎
𝟎 𝟎. 𝟐 𝟐 𝟎 𝟎
𝝁=( ) 𝐚𝐧𝐝 𝚺=[ ].
𝟎 𝟎 𝟎 𝟐 𝟎. 𝟐
𝟏 𝟎 𝟎 𝟎. 𝟐 𝟏
Then which one of the following statements is true?

(A) 5
[(𝑋1 + 𝑋2 )2 + (𝑋3 + 𝑋4 − 1)2 ] follows a central chi-square
17
distribution with 2 degrees of freedom

(B) 1 [(𝑋 + 𝑋 − 1)2 + (𝑋 + 𝑋 − 1)2 ] follows a central chi-square
3 1 3 2 4
distribution with 2 degrees of freedom

(C) 𝑋 +𝑋2 −1
𝐸 [√| 1 | ] is NOT finite
𝑋3 +𝑋4 −1

(D) 𝑋1 +𝑋2 +𝑋3 +𝑋4 −2
𝐸 [ | 𝑋 +𝑋 −𝑋 −𝑋 | ] is NOT finite
1 2 3 4

Q.43 Let 𝒀 follow 𝑵𝟖 (𝟎, 𝑰𝟖 ) distribution, where 𝑰𝟖 is the 𝟖 × 𝟖 identity matrix.
Let 𝒀𝑻 𝚺𝟏 𝒀 and 𝒀𝑻 𝚺𝟐 𝒀 be independent and follow central chi-square
distributions with 𝟑 and 𝟒 degrees of freedom, respectively, where 𝚺𝟏 and
𝚺𝟐 are 𝟖 × 𝟖 matrices and 𝒀𝑻 denotes transpose of 𝒀. Then which of the
following statements is/are true?
P : 𝚺𝟏 and 𝚺𝟐 are idempotent.
Q : 𝚺𝟏 𝚺𝟐 = 𝟎, where 𝟎 is the 𝟖 × 𝟖 zero matrix.

(A) P only

(B) Q only

(C) Both P and Q

(D) Neither P nor Q

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Q.44 – Q.55 Numerical Answer Type (NAT), carry TWO mark each (no negative marks).

Q.44 Let (𝑿, 𝒀) have a bivariate normal distribution with the joint probability
density function
𝟏 ( 𝟑 𝒙𝒚 − 𝟐𝟓 𝒙𝟐 − 𝟐 𝒚𝟐)
𝒇𝑿,𝒀 (𝒙, 𝒚) = 𝒆 𝟐 𝟑𝟐 , − ∞ < 𝒙, 𝒚 < ∞.
𝝅
Then 𝟖 𝑬(𝑿𝒀) equals __________

Q.45 Let 𝒇: ℝ × ℝ → ℝ be defined by 𝒇(𝒙, 𝒚) = 𝟖𝒙𝟐 − 𝟐𝒚, where ℝ denotes the
set of all real numbers. If 𝑴 and 𝒎 denote the maximum and minimum
values of 𝒇, respectively, on the set {(𝒙, 𝒚) ∈ ℝ × ℝ ∶ 𝒙𝟐 + 𝒚𝟐 = 𝟏}, then
𝑴 − 𝒎 equals __________ (round off to 𝟐 decimal places).

Q.46 Let 𝑨 = [𝒂 𝒖𝟏 𝒖𝟐 𝒖𝟑 ], 𝑩 = [𝒃 𝒖𝟏 𝒖𝟐 𝒖𝟑 ] and 𝑪 = [𝒖𝟐 𝒖𝟑 𝒖𝟏 𝒂 + 𝒃] be
three 𝟒 × 𝟒 real matrices, where 𝒂, 𝒃, 𝒖𝟏 , 𝒖𝟐 𝐚𝐧𝐝 𝒖𝟑 are 𝟒 × 𝟏 real column
vectors. Let 𝐝𝐞𝐭(𝑨), 𝐝𝐞𝐭(𝑩) and 𝐝𝐞𝐭(𝑪) denote the determinants of the
matrices 𝑨, 𝑩 and 𝑪, respectively. If 𝐝𝐞𝐭(𝑨) = 𝟔 and 𝐝𝐞𝐭(𝑩) = 𝟐, then
𝐝𝐞𝐭(𝑨 + 𝑩) − 𝐝𝐞𝐭(𝑪) equals __________

Q.47 Let 𝑿 be a random variable having the moment generating function
𝒆𝒕 − 𝟏
𝑴(𝒕) = , 𝒕 < 𝟏.
𝒕(𝟏 − 𝒕)
Then 𝑷(𝑿 > 𝟏) equals __________ (round off to 𝟐 decimal places).

Q.48 Let {𝑿𝒏 }𝒏≥𝟏 be a sequence of independent and identically distributed
random variables each having uniform distribution on [𝟎, 𝟑]. Let 𝒀 be a
random variable, independent of {𝑿𝒏 }𝒏≥𝟏 , having probability mass function
𝟏
, 𝒌 = 𝟏, 𝟐, … ,
𝑷(𝒀 = 𝒌) = {(𝒆 − 𝟏)𝒌!
𝟎, 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞.
Then 𝑷(𝐦𝐚𝐱{𝑿𝟏 , 𝑿𝟐 , … , 𝑿𝒀 } ≤ 𝟏) equals __________ (round off to 𝟐
decimal places).

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Q.49 Let {𝑿𝒏 }𝒏≥𝟏 be a sequence of independent and identically distributed
random variables each having probability density function
𝒆−𝒙 , 𝒙 > 𝟎,
𝒇(𝒙) = {
𝟎, 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞.
Let 𝑿(𝒏) = 𝐦𝐚𝐱{𝑿𝟏 , 𝑿𝟐 , … , 𝑿𝒏 } for 𝒏 ≥ 𝟏. If 𝒁 is the random variable to
which {𝑿(𝒏) − 𝐥𝐨𝐠 𝒆 𝒏}𝒏≥𝟏 converges in distribution, as 𝒏 → ∞, then the
median of
𝒁 equals __________ (round off to 𝟐 decimal places).

Q.50 Consider an amusement park where visitors are arriving according to a
Poisson process with rate 𝟏. Upon arrival, a visitor spends a random amount
of time in the park and then departs. The time spent by the visitors are
independent of one another, as well as of the arrival process, and have
common probability density function
𝒆−𝒙 , 𝒙 > 𝟎,
𝒇(𝒙) = {
𝟎, 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞.
If at a given time point, there are 𝟏𝟎 visitors in the park and 𝒑 is the
probability that there will be exactly two more arrivals before the next
departure, then
𝟏
equals __________
𝒑

Q.51 Let {𝟎. 𝟗𝟎, 𝟎. 𝟓𝟎, 𝟎. 𝟎𝟏, 𝟎. 𝟗𝟓} be a realization of a random sample of size 4
from the probability density function
𝜽 (𝟐𝜽−𝟏)⁄(𝟏−𝜽)
𝒇(𝒙) = { 𝟏 − 𝜽 𝒙 , 𝟎 < 𝒙 < 𝟏,
𝟎, 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞,
where 𝟎. 𝟓 ≤ 𝜽 < 𝟏. Then the maximum likelihood estimate of 𝜽 based on
the observed sample equals __________ (round off to 𝟐 decimal places).

Q.52 Let a random sample of size 𝟏𝟎𝟎 from a normal population with unknown
mean 𝝁 and variance 𝟗 give the sample mean 𝟓. 𝟔𝟎𝟖. Let 𝚽(⋅) denote
the distribution function of the standard normal random variable.
If 𝚽(𝟏. 𝟗𝟔) = 𝟎. 𝟗𝟕𝟓, 𝚽(𝟏. 𝟔𝟒) = 𝟎. 𝟗𝟓 and the uniformly most powerful
unbiased test based on sample mean is used to test 𝑯𝟎 : 𝝁 = 𝟓. 𝟎𝟐 against
𝑯𝟏 : 𝝁 ≠ 𝟓. 𝟎𝟐, then the 𝒑-value equals __________ (round off to 𝟑 decimal
places).

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Q.53 Let 𝑿 be a discrete random variable with probability mass function 𝒑 ∈
{𝒑𝟎 , 𝒑𝟏 }, where

𝒙 𝟕 𝟖 𝟗 𝟏𝟎

𝒑𝟏 (𝒙) 𝟎. 𝟔𝟗 𝟎. 𝟏𝟎 𝟎. 𝟏𝟔 𝟎. 𝟎𝟓

𝒑𝟎 (𝒙) 𝟎. 𝟗𝟎 𝟎. 𝟎𝟓 𝟎. 𝟎𝟒 𝟎. 𝟎𝟏

To test 𝑯𝟎 : 𝒑 = 𝒑𝟎 against 𝑯𝟏 : 𝒑 = 𝒑𝟏 , the power of the most powerful test
of size 𝟎. 𝟎𝟓, based on 𝑿, equals __________ (round off to 𝟐 decimal
places).

Q.54 Let 𝑿𝟏 , 𝑿𝟐 , … , 𝑿𝟏𝟎 be a random sample from a probability density function
𝒇𝜽 (𝒙) = 𝒇(𝒙 − 𝜽), −∞ < 𝒙 < ∞,
where −∞ < 𝜽 < ∞ and 𝒇(−𝒙) = 𝒇(𝒙) for −∞ < 𝒙 < ∞. For testing
𝑯𝟎 : 𝜽 = 𝟏. 𝟐 against 𝑯𝟏 : 𝜽 ≠ 𝟏. 𝟐, let 𝑻+ denote the Wilcoxson Signed-
rank test statistic. If 𝜼 denotes the probability of the event {𝑻+ < 𝟓𝟎} under
𝑯𝟎 , then 𝟑𝟐 𝜼 equals __________ (round off to 2 decimal places).

Q.55 Consider the multiple linear regression model
𝒀𝒊 = 𝜷𝟎 + 𝜷𝟏 𝒙𝟏,𝒊 + 𝜷𝟐 𝒙𝟐,𝒊 + ⋯ + 𝜷𝟐𝟐 𝒙𝟐𝟐,𝒊 + 𝝐𝒊 , 𝒊 = 𝟏, 𝟐, … , 𝟏𝟐𝟑,
where, for 𝒋 = 𝟎, 𝟏, 𝟐, … , 𝟐𝟐, 𝜷𝒋 ’s are unknown parameters and 𝝐𝒊 ’s are
independent and identically distributed 𝑵(𝟎, 𝝈𝟐 ), 𝝈 > 𝟎, random
variables.
If the sum of squares due to regression is 𝟑𝟑𝟖. 𝟗𝟐, the total sum of squares
is 𝟓𝟐𝟐. 𝟑𝟎 and 𝑹𝟐𝒂𝒅𝒋 denotes the value of adjusted 𝑹𝟐 , then 𝟏𝟎𝟎 𝑹𝟐𝒂𝒅𝒋
equals __________ (round off to 𝟐 decimal places).

END OF THE QUESTION PAPER

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