JAM 2024 Question Paper Mathematical Statistics (MS)

JOINT ADMISSION TEST FOR MASTERS

2024

IIT

JAM
2024
Question Papers | Answer Key

The Joint Admission Test for Masters is a
common admission test conducted every
year for admission into Master of Science
and other post-graduate science
programs at Indian Institutes of
Technology

JAM 2024 Mathematical Statistics (MS)

Notations and Useful Data

ℕ The set of positive integers
ℝ The set of real numbers
ℝ𝑛 {(𝑥1 , 𝑥2 , … , 𝑥𝑛 ): 𝑥𝑖 ∈ ℝ, 𝑖 = 1, 2, … , 𝑛}, 𝑛 = 2, 3, …
ln 𝑥 Natural logarithm of 𝑥, 𝑥 > 0
det(𝑀) Determinant of a square matrix 𝑀
𝑎𝑑𝑗 𝑀 Adjoint of a square matrix 𝑀, that is, transpose of cofactor matrix of 𝑀
∅ Empty set
𝐸𝐶 Complement of event 𝐸
𝑃(𝐸) Probability of event 𝐸
𝑃(𝐸|𝐹) Conditional probability of event 𝐸 given the occurrence of event 𝐹
𝐸(𝑋) Expectation of a random variable 𝑋
𝑉𝑎𝑟(𝑋) Variance of a random variable 𝑋
𝐶𝑜𝑣(𝑋, 𝑌) Covariance between random variables 𝑋 and 𝑌
𝐵𝑖𝑛(𝑛, 𝑝) Binomial distribution with parameters 𝑛 and 𝑝, 𝑛 ∈ ℕ, 0 < 𝑝 < 1
𝑈(𝑎, 𝑏) Continuous uniform distribution on the interval (𝑎, 𝑏), 𝑎 < 𝑏, 𝑎, 𝑏 ∈ ℝ
𝐸𝑥𝑝(𝜆) Exponential distribution with the probability density function
−𝜆𝑥
𝑓(𝑥) = {𝜆 𝑒 , if 𝑥 > 0 , for 𝜆 > 0.
0 , if 𝑥 ≤ 0
2
𝑁(𝜇, 𝜎 ) Normal distribution with mean 𝜇 and variance 𝜎 2 , 𝜇 ∈ ℝ , 𝜎 > 0
𝑁2 (𝜇1 , 𝜇2 , 𝜎12 , 𝜎22 , 𝜌) Bivariate normal distribution with means 𝜇1 , 𝜇2 , variances 𝜎12 , 𝜎22 and
correlation 𝜌, 𝜇1 ∈ ℝ, 𝜇2 ∈ ℝ, 𝜎1 > 0, 𝜎2 > 0, −1 < 𝜌 < 1
𝜙(⋅) The probability density function of 𝑁(0, 1) random variable
Φ(⋅) The cumulative distribution function of 𝑁(0,1) random variable
2
𝜒𝑛 Central chi-square distribution with 𝑛 degrees of freedom, 𝑛 = 1,2, …
𝑡𝑛 Central Student’s 𝑡 distribution with 𝑛 degrees of freedom, 𝑛 = 1,2, …
𝐹𝑚,𝑛 Snedecor’s central 𝐹-distribution with 𝑚 and 𝑛 degrees of freedom,
𝑚, 𝑛 ∈ ℕ
2 2
𝜒𝑛,𝛼 A constant such that 𝑃(𝑋 > 𝜒𝑛,𝛼 ) = 𝛼, where 𝑋 has central chi-square
distribution with 𝑛 degrees of freedom, 𝑛 = 1,2, … ; 𝛼 ∈ (0, 1)
𝑡𝑛,𝛼 A constant such that 𝑃(𝑋 > 𝑡𝑛,𝛼 ) = 𝛼, where 𝑋 has central Student’s 𝑡
distribution with 𝑛 degrees of freedom, 𝑛 = 1,2, … ; 𝛼 ∈ (0, 1)
→
𝑑 Convergence in distribution
𝑃
→ Convergence in probability
i. i. d. Independent and identically distributed

MS 1/41

JAM 2024 Mathematical Statistics (MS)

Section A: Q.1 – Q.10 Carry ONE mark each.

Q.1 1 1 𝑛2
Let 𝑎𝑛 = 1 + 2 + ⋯ + 𝑛 and 𝑏𝑛 = 2𝑛 for all 𝑛 ∈ ℕ. Then

(A) {𝑎𝑛 } is a Cauchy sequence but {𝑏𝑛 } is NOT a Cauchy sequence

(B) {𝑎𝑛 } is NOT a Cauchy sequence but {𝑏𝑛 } is a Cauchy sequence

(C) both {𝑎𝑛 } and {𝑏𝑛 } are Cauchy sequences

(D) neither {𝑎𝑛 } nor {𝑏𝑛 } is a Cauchy sequence

Q.2 Let 𝑓(𝑥, 𝑦) = 2𝑥 4 − 3𝑦 2 for all (𝑥, 𝑦) ∈ ℝ2 . Then

(A) 𝑓 has a point of local minimum

(B) 𝑓 has a point of local maximum

(C) 𝑓 has a saddle point

(D) 𝑓 has no point of local minimum, no point of local maximum, and no saddle point

MS 2/41

JAM 2024 Mathematical Statistics (MS)

Q.3 Let 𝐴 = (
𝑎 0
) be a real matrix, where 𝑎𝑑 = 1 and 𝑐 ≠ 0. If
𝑐 𝑑
𝛼 𝛽
𝐴−1 + (𝑎𝑑𝑗 𝐴)−1 = ( ),
𝛾 𝛿
then (𝛼, 𝛽, 𝛾, 𝛿) is equal to

(A) (𝑎 + 𝑑, 0, 0, 𝑎 + 𝑑)

(B) (𝑎 + 𝑑, 0, 𝑐, 𝑎 + 𝑑)

(C) (𝑎, 0, 0, 𝑑 )

(D) (𝑎, 0, 𝑐, 𝑑 )

MS 3/41

JAM 2024 Mathematical Statistics (MS)

Q.4 A bag has 5 blue balls and 15 red balls. Three balls are drawn at random from the
bag simultaneously. Then the probability that none of the chosen balls is blue
equals

(A) 75
152

(B) 91
228

(C) 27
64

(D) 273
800

Q.5 Let 𝑌 be a continuous random variable such that 𝑃(𝑌 > 0) = 1 and 𝐸(𝑌) = 1.
For 𝑝 ∈ (0,1), let 𝜉𝑝 denote the 𝑝th quantile of the probability distribution of the
random variable 𝑌. Then which of the following statements is always correct?

(A) 𝜉0.75 ≥ 5

(B) 𝜉0.75 ≤ 4

(C) 𝜉0.25 ≥ 4

(D) 𝜉0.25 = 2

MS 4/41

JAM 2024 Mathematical Statistics (MS)

Q.6 Let 𝑋 be a continuous random variable having the 𝑈(−2, 3) distribution. Then
which of the following statements is correct?

(A) 2𝑋 + 5 has the 𝑈(1, 10) distribution

(B) 7 − 6𝑋 has the 𝑈(−11, 19) distribution

(C) 3𝑋 2 + 5 has the 𝑈(5, 32) distribution

(D) |𝑋| has the 𝑈(0, 3) distribution

Q.7 Let 𝑋 be a random variable having the Poisson distribution with mean 1. Let
𝑔: ℕ ∪ {0} → ℝ be defined by

1, if 𝑥 ∈ {0,2}
𝑔(𝑥) = { .
0, if 𝑥 ∉ {0,2}

Then 𝐸(𝑔(𝑋)) is equal to

(A) 𝑒 −1

(B) 2𝑒 −1

(C) 5
𝑒 −1
2

(D) 3
𝑒 −1
2

MS 5/41

JAM 2024 Mathematical Statistics (MS)

Q.8 For 𝑛 ∈ ℕ, let 𝑍𝑛 be the smallest order statistic based on a random sample of
𝑑
size 𝑛 from the 𝑈(0,1) distribution. Let 𝑛𝑍𝑛 → 𝑍, as 𝑛 → ∞, for some random
variable 𝑍. Then 𝑃(𝑍 ≤ ln 3) is equal to

(A) 1
4

(B) 2
3

(C) 3
4

(D) 1
3

Q.9 Let 𝑋1 , 𝑋2 , … , 𝑋20 be a random sample from the 𝑁(5, 2) distribution and let
1
𝑌𝑖 = 𝑋2𝑖 − 𝑋2𝑖−1 , 𝑖 = 1,2, … ,10. Then 𝑊 = 4 ∑10 2
𝑖=1 𝑌𝑖 has the

(A) 𝑡20 distribution

2
(B) 𝜒20 distribution

2
(C) 𝜒10 distribution

(D) 𝑁(250, 20) distribution

MS 6/41

JAM 2024 Mathematical Statistics (MS)

Q.10 Let 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 be the observed values of a random sample from a 𝑁(𝜇, 𝜎 2 )
distribution, where 𝜇 ∈ ℝ and 𝜎 ∈ (0, ∞) are unknown parameters. Let 𝑥̅ and
1
𝑠 = √3 ∑4𝑖=1(𝑥𝑖 − 𝑥̅ )2 be the observed sample mean and the sample standard

deviation, respectively. For testing 𝐻0 : 𝜇 = 0 against 𝐻1 : 𝜇 ≠ 0, the likelihood
|𝑥̅ |
ratio test of size 𝛼 = 0.05 rejects 𝐻0 if and only if > 𝑘. Then the value of 𝑘
𝑠

is

(A) 1
𝑡
2 3,0.025

(B) 𝑡3,0.025

(C) 2𝑡3,0.05

(D) 1
𝑡
2 3,0.05

MS 7/41

JAM 2024 Mathematical Statistics (MS)

Section A: Q.11 – Q.30 Carry TWO marks each.

Q.11 1 1
For 𝑛 ∈ ℕ, let 𝑎𝑛 = √𝑛 sin2 (𝑛) cos 𝑛, and 𝑏𝑛 = √𝑛 sin (𝑛2 ) cos 𝑛. Then

(A) the series ∑∞ ∞
𝑛=1 𝑎𝑛 converges but the series ∑𝑛=1 𝑏𝑛 does NOT converge

(B) the series ∑∞ ∞
𝑛=1 𝑎𝑛 does NOT converge but the series ∑𝑛=1 𝑏𝑛 converges

(C) both the series ∑∞ ∞
𝑛=1 𝑎𝑛 and ∑𝑛=1 𝑏𝑛 converge

(D) neither the series ∑∞ ∞
𝑛=1 𝑎𝑛 nor the series ∑𝑛=1 𝑏𝑛 converges

MS 8/41

JAM 2024 Mathematical Statistics (MS)

Q.12 Let 𝑓𝑖 : ℝ → ℝ, 𝑖 = 1,2, be defined by

1 1
( ) ( )
𝑓1 (𝑥) = {sin 𝑥 + cos 𝑥 , if 𝑥 ≠ 0
0, if 𝑥 = 0
and
1 1
(𝑥) 𝑥 (sin ( ) + cos ( ) ) , if 𝑥 ≠ 0 .
𝑓2 ={ 𝑥 𝑥
0, if 𝑥 = 0

Then

(A) 𝑓1 is continuous at 0 but 𝑓2 is NOT continuous at 0

(B) 𝑓1 is NOT continuous at 0 but 𝑓2 is continuous at 0

(C) both 𝑓1 and 𝑓2 are continuous at 0

(D) neither 𝑓1 nor 𝑓2 is continuous at 0

MS 9/41

JAM 2024 Mathematical Statistics (MS)

Q.13 Let 𝑓(𝑥, 𝑦) = |𝑥𝑦| + 𝑥 for all (𝑥, 𝑦) ∈ ℝ2 . Then the partial derivative of 𝑓 with
respect to 𝑥 exists

(A) at (0,0) but NOT at (0,1)

(B) at (0,1) but NOT at (0,0)

(C) at (0,0) and (0,1), both

(D) neither at (0,0) nor at (0,1)

Q.14 Let 𝑓(𝑥) = 4𝑥 2 − sin 𝑥 + cos 2𝑥 for all 𝑥 ∈ ℝ. Then 𝑓 has

(A) a point of local maximum

(B) no point of local minimum

(C) exactly one point of local minimum

(D) at least two points of local minima

MS 10/41

JAM 2024 Mathematical Statistics (MS)

Q.15 Consider the improper integrals
∞ ∞
𝑡 sin 𝑡 1 1
𝐼1 = ∫ 𝑡
𝑑𝑡 and 𝐼2 = ∫ ln (1 + ) 𝑑𝑡.
1 𝑒 1 √𝑡 𝑡
Then

(A) 𝐼1 converges but 𝐼2 does NOT converge

(B) 𝐼1 does NOT converge but 𝐼2 converges

(C) both 𝐼1 and 𝐼2 converge

(D) neither 𝐼1 nor 𝐼2 converges

MS 11/41

JAM 2024 Mathematical Statistics (MS)

Q.16 Let 𝐴 be a 3 × 5 matrix defined by
0 1 3 1 2
𝐴 = (1 6 2 3 4) .
1 8 8 5 8
Consider the system of linear equations given by

𝑥1
𝑥2 3
𝐴 𝑥3 = ( 1 ) ,
𝑥4 10
𝑥
( )
5

where 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 are real variables. Then

(A) the rank of 𝐴 is 2 and the given system has a solution

(B) the rank of 𝐴 is 2 and the given system does NOT have a solution

(C) the rank of 𝐴 is 3 and the given system has a solution

(D) the rank of 𝐴 is 3 and the given system does NOT have a solution

MS 12/41

JAM 2024 Mathematical Statistics (MS)

Q.17 Let Ω = {1,2,3,4,5,6}. Then which of the following classes of sets is an algebra?

(A) ℱ1 = {∅, Ω, {1,2}, {3,4}, {3,6}}

(B) ℱ2 = {∅, Ω, {1,2,3}, {4,5,6}}

(C) ℱ3 = {∅, Ω, {1,2}, {4,5}, {1,2,4,5}, {3,4,5,6}, {1,2,3,6}}

(D) ℱ4 = {∅, {4,5}, {1,2,3,6}}

Q.18 Two fair coins 𝑆1 and 𝑆2 are tossed independently once. Let the events 𝐸, 𝐹 and
𝐺 be defined as follows:

𝐸: Head appears on 𝑆1

𝐹: Head appears on 𝑆2

𝐺: The same outcome (head or tail) appears on both 𝑆1 and 𝑆2

Then which of the following statements is NOT correct?

(A) 𝐸 and 𝐹 are independent

(B) 𝐹 and 𝐺 are independent

(C) 𝐸 and 𝐺 𝐶 are independent

(D) 𝐸, 𝐹, and 𝐺 are mutually independent

MS 13/41

JAM 2024 Mathematical Statistics (MS)

Q.19 Let 𝑓1 (𝑥) be the probability density function of the 𝑁(0,1) distribution and 𝑓2 (𝑥)
be the probability density function of the 𝑁(0, 6) distribution. Let 𝑌 be a random
variable with probability density function

𝑓(𝑥) = 0.6 𝑓1 (𝑥) + 0.4 𝑓2 (𝑥) , − ∞ < 𝑥 < ∞.
Then 𝑉𝑎𝑟(𝑌) is equal to

(A) 7

(B) 3

(C) 3.5

(D) 1

MS 14/41

JAM 2024 Mathematical Statistics (MS)

Q.20 Which of the following functions represents a cumulative distribution function?

𝜋
(A) 0, if 𝑥 < 4
𝜋 3𝜋
𝐹1 (𝑥) = sin 𝑥 , if 4 ≤ 𝑥 < 4
3𝜋
1, if 𝑥 ≥ 4
{

(B) 0, if 𝑥 < 0
𝜋
𝐹 (𝑥) = {2 sin 𝑥 , if 0 ≤ 𝑥 < 4
2
𝜋
1, if 𝑥 ≥ 4

(C) 0, if 𝑥 < 0
1
𝑥, if 0 ≤ 𝑥 < 3
𝐹3 (𝑥) = 1
1 1
𝑥+3 , if 3 ≤ 𝑥 ≤ 2
1
{ 1, if 𝑥 > 2

(D) 0, if 𝑥 < 0
𝜋
𝐹 (𝑥) = {√2 sin 𝑥 ,
4
if 0 ≤ 𝑥 < 4
𝜋
1, if 𝑥 ≥ 4

MS 15/41

JAM 2024 Mathematical Statistics (MS)

Q.21 Let 𝑋 be a random variable such that 𝑋 and −𝑋 have the same distribution.
Let 𝑌 = 𝑋 2 be a continuous random variable with the probability density function

𝑦
𝑒− 2
, if 𝑦 > 0
𝑔(𝑦) = {√2𝜋𝑦 .
0, if 𝑦 ≤ 0

Then 𝐸((𝑋 − 1)4 ) is equal to

(A) 9

(B) 10

(C) 11

(D) 12

MS 16/41

JAM 2024 Mathematical Statistics (MS)

Q.22 1
Suppose that random variable 𝑋 has 𝐸𝑥𝑝 (5) distribution and, for any 𝑥 > 0, the

conditional distribution of random variable 𝑌, given 𝑋 = 𝑥, is 𝑁(𝑥, 2). Then
𝑉𝑎𝑟(𝑋 + 𝑌) is equal to

(A) 52

(B) 50

(C) 2

(D) 102

MS 17/41

JAM 2024 Mathematical Statistics (MS)

Q.23 Let the random vector (𝑋, 𝑌) have the joint probability density function

1
, if 0 < 𝑦 < 𝑥 < 1
𝑓(𝑥, 𝑦) = {𝑥 .
0, otherwise
Then 𝐶𝑜𝑣(𝑋, 𝑌) is equal to

(A) 1
6

(B) 1
12

(C) 1
18

(D) 1
24

MS 18/41

JAM 2024 Mathematical Statistics (MS)

Q.24 Let (𝑋1 , 𝑌1 ), (𝑋2 , 𝑌2 ), … , (𝑋20 , 𝑌20 ) be a random sample from the
3 1 1
𝑁2 (0, 0, 1, 1, 4 ) distribution. Define 𝑋̅ = 20 ∑20 ̅ 20
𝑖=1 𝑋𝑖 and 𝑌 = 20 ∑𝑖=1 𝑌𝑖 . Then

𝑉𝑎𝑟(𝑋̅ − 𝑌̅) is equal to

(A) 1
16

(B) 1
40

(C) 1
10

(D) 3
40

MS 19/41

JAM 2024 Mathematical Statistics (MS)

Q.25 1
For 𝑛 ∈ ℕ, let 𝑋𝑛 be a random variable having the 𝐵𝑖𝑛 (𝑛, 4) distribution. Then

2𝑛 − √3𝑛 𝑛 𝑛
lim [𝑃 (𝑋𝑛 ≤ ) + 𝑃 ( ≤ 𝑋𝑛 ≤ )]
𝑛→∞ 8 6 3

is equal to

(You may use Φ(0.5) = 0.6915, Φ(1) = 0.8413, Φ(1.5) = 0.9332, Φ(2) =
0.9772 )

(A) 1.6915

(B) 1.3085

(C) 1.1587

(D) 0.6915

MS 20/41

JAM 2024 Mathematical Statistics (MS)

Q.26 Let 𝑋1 , 𝑋2 , … , 𝑋10 be a random sample from the 𝑁(3,4) distribution and let
𝑌1 , 𝑌2 , … , 𝑌15 be a random sample from the 𝑁(−3,6) distribution. Assume that
the two samples are drawn independently. Define

1 1 1 10
𝑋̅ = 10 ∑10 ̅ 15 ̅ 2
𝑖=1 𝑋𝑖 , 𝑌 = 15 ∑𝑗=1 𝑌𝑗 , and 𝑆 = √9 ∑𝑖=1(𝑋𝑖 − 𝑋 ) .

√5(𝑋̅+ 𝑌̅)
Then the distribution of 𝑈 = is
𝑆

(A) 𝑁 (0, 4)
5

(B) 𝜒92

(C) 𝑡9

(D) 𝑡23

MS 21/41

JAM 2024 Mathematical Statistics (MS)

Q.27 For 𝑛 ≥ 2, let 𝜖1 , 𝜖2 , … , 𝜖𝑛 be i.i.d. random variables having the 𝑁(0,1)
distribution. Consider 𝑛 independent random variables 𝑌1 , 𝑌2 , … , 𝑌𝑛 defined by

𝑌𝑖 = 𝛽𝑖 + 𝜖𝑖 , 𝑖 = 1,2, … , 𝑛 ,
1 2𝑌̅ 1 𝑌
where 𝛽 ∈ ℝ. Define 𝑌̅ = 𝑛 ∑𝑛𝑖=1 𝑌𝑖 , 𝑇1 = 𝑛+1 , and 𝑇2 = 𝑛 ∑𝑛𝑖=1 𝑖𝑖 . Then which

of the following statements is NOT correct?

(A) 𝑇1 is an unbiased estimator of 𝛽

(B) 𝑇2 is an unbiased estimator of 𝛽

(C) 𝑉𝑎𝑟(𝑇1 ) < 𝑉𝑎𝑟(𝑇2 )

(D) 𝑉𝑎𝑟(𝑇1 ) = 𝑉𝑎𝑟(𝑇2 )

MS 22/41

JAM 2024 Mathematical Statistics (MS)

Q.28 A biased coin, with probability of head as 𝑝, is tossed 𝑚 times independently. It
1 3
is known that 𝑝 ∈ {4 , 4 } and 𝑚 ∈ {3 , 5}. If 3 heads are observed in these 𝑚

tosses, then which of the following statements is correct?

(A) (3, 3) is a maximum likelihood estimator of (𝑚, 𝑝)
4

(B) (5, 1) is a maximum likelihood estimator of (𝑚, 𝑝)
4

(C) (5, 3) is a maximum likelihood estimator of (𝑚, 𝑝)
4

(D) Maximum likelihood estimator of (𝑚, 𝑝) is NOT unique

MS 23/41

JAM 2024 Mathematical Statistics (MS)

Q.29 Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 be a random sample from an 𝐸𝑥𝑝(𝜆) distribution, where
𝜆 ∈ {1, 2}. For testing 𝐻0 : 𝜆 = 1 against 𝐻1 : 𝜆 = 2 , the most powerful test of
size 𝛼, 𝛼 ∈ (0,1), will reject 𝐻0 if and only if

(A) ∑𝑛𝑖=1 𝑋𝑖 ≤ 1 𝜒2𝑛,1−𝛼
2
2

(B) ∑𝑛𝑖=1 𝑋𝑖 ≥ 2 𝜒2𝑛,1−𝛼
2

(C) ∑𝑛𝑖=1 𝑋𝑖 ≤ 1 𝜒𝑛,1−𝛼
2
2

(D) ∑𝑛𝑖=1 𝑋𝑖 ≥ 2 𝜒𝑛,1−𝛼
2

MS 24/41

JAM 2024 Mathematical Statistics (MS)

Q.30 Let 𝑋1 , 𝑋2 , … , 𝑋10 be a random sample from a 𝑁(0, 𝜎 2 ) distribution, where
σ > 0 is unknown. For testing 𝐻0 : 𝜎 2 ≤ 1 against 𝐻1 : 𝜎 2 > 1, a test of size
𝛼 = 0.05 rejects 𝐻0 if and only if ∑10 2
𝑖=1 𝑋𝑖 > 18.307. Let 𝛽 be the power of this

test, at 𝜎 2 = 2. Then 𝛽 lies in the interval

2 2 2
(You may use 𝜒10,0.05 = 18.307, 𝜒10,0.1 = 15.9872, 𝜒10,0.25 = 12.5489,
2 2 2 2
𝜒10,0.5 = 9.3418, 𝜒10,0.75 = 6.7372, 𝜒10,0.9 = 4.8652, 𝜒10,0.95 = 3.9403,
2
𝜒10,0.975 = 3.247 )

(A) (0.50, 0.75)

(B) (0.75, 0.90)

(C) (0.90, 0.95)

(D) (0.95, 0.975)

MS 25/41

JAM 2024 Mathematical Statistics (MS)

Section B: Q.31 – Q.40 Carry TWO marks each.

Q.31 Let 𝑎1 = 1, 𝑎𝑛+1 = 𝑎𝑛 (
√𝑛+sin 𝑛
) and 𝑏𝑛 = 𝑎𝑛2 for all 𝑛 ∈ ℕ. Then which of the
𝑛

following statements is/are correct?

(A) the series ∑∞
𝑛=1 𝑎𝑛 converges

(B) the series ∑∞
𝑛=1 𝑏𝑛 converges

(C) the series ∑∞ ∞
𝑛=1 𝑎𝑛 converges but the series ∑𝑛=1 𝑏𝑛 does NOT converge

(D) neither the series ∑∞ ∞
𝑛=1 𝑎𝑛 nor the series ∑𝑛=1 𝑏𝑛 converges

MS 26/41

JAM 2024 Mathematical Statistics (MS)

Q.32 Let 𝑓: ℝ → ℝ be a twice differentiable function such that

𝑓(0) = 0, 𝑓(2) = 4, 𝑓(4) = 4 and 𝑓(8) = 12.

Then which of the following statements is/are correct?

(A) 𝑓 ′ (𝑥) ≤ 1 for all 𝑥 ∈ [0, 2]

(B) 𝑓 ′ (𝑥1 ) > 1 for some 𝑥1 ∈ [0, 2]

(C) 𝑓 ′ (𝑥2 ) > 1 for some 𝑥2 ∈ [4, 8]

(D) 𝑓 ′′ (𝑥3 ) = 0 for some 𝑥3 ∈ [0, 8]

Q.33 Let 𝐴 be a 3 × 3 real matrix. Suppose that 1 and 2 are characteristic roots of 𝐴,
and 12 is a characteristic root of 𝐴 + 𝐴2 . Then which of the following statements
is/are correct?

(A) det(𝐴) ≠ 0

(B) det(𝐴 + 𝐴2 ) ≠ 0

(C) det(𝐴) = 0

(D) trace of (𝐴 + 𝐴2 ) is 20

MS 27/41

JAM 2024 Mathematical Statistics (MS)

Q.34 Consider four dice 𝐷1 , 𝐷2 , 𝐷3 , and 𝐷4 , each having six faces marked as follows:

Die Marks on faces

𝐷1 4, 4, 4, 4, 0, 0

𝐷2 3, 3, 3, 3, 3, 3

𝐷3 6, 6, 2, 2, 2, 2

𝐷4 5, 5, 5, 1, 1, 1

In each roll of a die, each of its six faces is equally likely to occur. Suppose that
each of these four dice is rolled once, and the marks on their upper faces are
noted. Let the four rolls be independent. If 𝑋𝑖 denotes the mark on the upper face
of the die 𝐷𝑖 , 𝑖 = 1, 2, 3, 4, then which of the following statements is/are correct?

(A) 𝑃(𝑋1 > 𝑋2 ) = 2
3

(B) 𝑃(𝑋3 > 𝑋4 ) = 2
3

(C) 𝑃(𝑋2 > 𝑋3 ) = 1
3

(D) The events {𝑋1 > 𝑋2 } and {𝑋2 > 𝑋3 } are independent

MS 28/41

JAM 2024 Mathematical Statistics (MS)

Q.35 Let 𝑋 be a continuous random variable with a probability density function 𝑓 and
the moment generating function 𝑀(𝑡). Suppose that 𝑓(𝑥) = 𝑓(−𝑥) for all
𝑥 ∈ ℝ and the moment generating function 𝑀(𝑡) exists for 𝑡 ∈ (−1, 1). Then
which of the following statements is/are correct?

(A) 𝑃(𝑋 = −𝑋) = 1

(B) 0 is the median of 𝑋

(C) 𝑀(𝑡) = 𝑀(−𝑡) for all 𝑡 ∈ (−1, 1)

(D) 𝐸(𝑋) = 1

Q.36 Let 𝑋 and 𝑌 be independent random variables having 𝐵𝑖𝑛(18, 0.5) and
𝐵𝑖𝑛(20, 0.5) distributions, respectively. Further, let 𝑈 = min{𝑋, 𝑌} and
𝑉 = max{𝑋, 𝑌}. Then which of the following statements is/are correct?

(A) 𝐸(𝑈 + 𝑉) = 19

(B) 𝐸(|𝑋 − 𝑌|) = 𝐸(𝑉 − 𝑈)

(C) 𝑉𝑎𝑟(𝑈 + 𝑉) = 16

(D) 38 − (𝑋 + 𝑌) has 𝐵𝑖𝑛(38, 0.5) distribution

MS 29/41

JAM 2024 Mathematical Statistics (MS)

Q.37 Let 𝑋 and 𝑌 be continuous random variables having the joint probability density
function

𝑒 −𝑥 , if 0 ≤ 𝑦 < 𝑥 < ∞
𝑓(𝑥, 𝑦) = { .
0, otherwise

Then which of the following statements is/are correct?

(A) 𝑃(𝑌 2 = 3𝑋) = 0

(B) 𝑃(𝑋 > 2𝑌) = 1
2

(C) 𝑃(𝑋 − 𝑌 ≥ 1) = 𝑒 −1

(D) 𝑃(𝑋 > ln 2 | 𝑌 > ln 3) = 0

MS 30/41

JAM 2024 Mathematical Statistics (MS)

Q.38 For 𝑛 ≥ 2, let 𝑋1 , 𝑋2 , … , 𝑋𝑛 be a random sample from a distribution with
𝐸(𝑋1 ) = 0 , 𝑉𝑎𝑟(𝑋1 ) = 1 and 𝐸(𝑋14 ) < ∞. Let

𝑛 𝑛
1 1
𝑋̅𝑛 = ∑ 𝑋𝑖 and 𝑆𝑛2 = ∑(𝑋𝑖 − 𝑋̅𝑛 )2 .
𝑛 𝑛−1
𝑖=1 𝑖=1

Then which of the following statements is/are always correct?

(A) 𝐸(𝑆𝑛2 ) = 1 for all 𝑛 ≥ 2

(B) 𝑑
√𝑛 𝑋̅𝑛 → 𝑍 as 𝑛 → ∞, where 𝑍 has the 𝑁(0, 1) distribution

(C) 𝑋̅𝑛 and 𝑆𝑛2 are independently distributed for all 𝑛 ≥ 2

(D) 1 𝑃
∑𝑛𝑖=1 𝑋𝑖2 → 2, as 𝑛 → ∞
𝑛

MS 31/41

JAM 2024 Mathematical Statistics (MS)

Q.39 Let 𝑋1 , 𝑋2 , … , 𝑋50 be a random sample from a 𝑁(0, 𝜎 2 ) distribution, where
𝜎 > 0. Define

25 25
1 1
𝑋̅𝑒 = ∑ 𝑋2𝑖 , 𝑋̅𝑜 = ∑ 𝑋2𝑖−1 ,
25 25
𝑖=1 𝑖=1

25 25
1 1
𝑆𝑒 = √ ∑(𝑋2𝑖 − 𝑋̅𝑒 )2 and 𝑆𝑜 = √ ∑(𝑋2𝑖−1 − 𝑋̅𝑜 )2 .
24 24
𝑖=1 𝑖=1

Then which of the following statements is/are correct?

(A) 5𝑋̅𝑒
has 𝑡24 distribution
𝑆𝑒

(B) 5(𝑋̅𝑒 +𝑋̅𝑜 )
has 𝑡49 distribution
√𝑆𝑒2 +𝑆𝑜2

(C) 49𝑆𝑜2 2
has 𝜒49 distribution
𝜎2

(D) 𝑆𝑜2
has 𝐹24,24 distribution
𝑆𝑒2

MS 32/41

JAM 2024 Mathematical Statistics (MS)

Q.40 Let 𝜃0 and 𝜃1 be real constants such that 𝜃1 > 𝜃0 . Suppose that a random sample
is taken from a 𝑁(𝜃, 1) distribution, 𝜃 ∈ ℝ. For testing 𝐻0 : 𝜃 = 𝜃0 against
𝐻1 : 𝜃 = 𝜃1 at level 0.05, let 𝛼 and 𝛽 denote the size and the power, respectively,
of the most powerful test, 𝜓0 . Then which of the following statements is/are
correct?

(A) 𝛽 < 𝛼

(B) The test 𝜓0 is the uniformly most powerful test of level 𝛼 for testing 𝐻0 : 𝜃 = 𝜃0
against 𝐻1 : 𝜃 > 𝜃0

(C) 𝛼 < 𝛽

(D) The test 𝜓0 is the uniformly most powerful test of level 𝛼 for testing 𝐻0 : 𝜃 = 𝜃0
against 𝐻1 : 𝜃 < 𝜃0

MS 33/41

JAM 2024 Mathematical Statistics (MS)

Section C: Q.41 – Q.50 Carry ONE mark each.

Q.41 2𝑛(𝑥+3)𝑛
The radius of convergence of the power series ∑∞
𝑛=0 is equal to
5𝑛

__________ (answer in integer)

Q.42 Let 𝑓(𝑥) = ∫−1
𝑥 2 −2𝑥 2
𝑒 𝑡 −𝑡 𝑑𝑡 for all 𝑥 ∈ ℝ. If 𝑓 is decreasing on (0, 𝑚) and
increasing on (𝑚, ∞), then the value of 𝑚 is equal to __________ (answer in
integer)

Q.43 Let 𝑉 = {(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) ∈ ℝ4 ∶ 𝑥1 = 𝑥2 }. Consider 𝑉 as a subspace of ℝ4 over
the real field. Then the dimension of 𝑉 is equal to __________ (answer in integer)

MS 34/41

JAM 2024 Mathematical Statistics (MS)

Q.44 If 12 fair dice are independently rolled, then the probability of obtaining at least
two sixes is equal to __________ (round off to 2 decimal places)

Q.45 Let 𝑋 be a random variable with the moment generating function

(1 + 3𝑒 𝑡 )2
𝑀(𝑡) = , −∞ < 𝑡 < ∞ .
16

Let 𝛼 = 𝐸(𝑋) − 𝑉𝑎𝑟(𝑋). Then the value of 8𝛼 is equal to __________
(answer in integer)

Q.46 For 𝑛 ∈ ℕ, let 𝑋1 , 𝑋2 , … , 𝑋𝑛 be a random sample from the Cauchy distribution
having probability density function

1
𝑓(𝑥) = , −∞ < 𝑥 < ∞.
𝜋(1 + 𝑥 2 )

Let 𝑔: ℝ → ℝ be defined by
𝑥, if − 1000 ≤ 𝑥 ≤ 1000
𝑔(𝑥) = { .
0, otherwise

Let
𝑛
1 1
𝛼 = lim 𝑃 ( 3 ∑ 𝑔(𝑋𝑖 ) > ).
𝑛→∞
𝑛4 𝑖=1 2

Then 100𝛼 is equal to __________ (answer in integer)

MS 35/41

JAM 2024 Mathematical Statistics (MS)

Q.47 For 𝑛 ∈ ℕ, let 𝑋1 , 𝑋2 , … , 𝑋𝑛 be a random sample from the 𝐹20,40 distribution.
1 1
Then, as 𝑛 → ∞, 𝑛 ∑𝑛𝑖=1 𝑋 converges in probability to __________
𝑖

(round off to 2 decimal places)

Q.48 Let 𝑋1 , 𝑋2 , … , 𝑋10 be a random sample from the 𝐸𝑥𝑝(1) distribution. Define
𝑊 = max{𝑒 −𝑋1 , 𝑒 −𝑋2 , … , 𝑒 −𝑋10 }. Then the value of 22𝐸(𝑊) is equal to

__________ (answer in integer)

Q.49 Let 𝑋1 , 𝑋2 , 𝑋3 be i.i.d. random variables from a continuous distribution having
probability density function
1 1
3
, if 𝑥 >
𝑓(𝑥) = {2𝑥 2 .
1
0, if 𝑥 ≤
2

Let 𝑋(1) = min{𝑋1 , 𝑋2 , 𝑋3 }. Then the value of 10𝐸(𝑋(1) ) is equal to
__________ (answer in integer)

MS 36/41

JAM 2024 Mathematical Statistics (MS)

Q.50 Suppose that the lifetimes (in months) of bulbs manufactured by a company have
an 𝐸𝑥𝑝(𝜆) distribution, where 𝜆 > 0. A random sample of size 10 taken from the
bulbs manufactured by the company yields the sample mean lifetime 𝑥̅ = 3.52
1
months. Then the uniformly minimum variance unbiased estimate of 𝜆 based on

this sample is equal to __________ months (round off to 2 decimal places)

Section C: Q.51 – Q.60 Carry TWO marks each.

Q.51 1 1 1
1
The value of lim 𝑛 (sin 2𝑛 − 2 𝑒 −𝑛 + 2) is equal to __________ (answer in
𝑛→∞

integer)

Q.52 The value of the integral
2 √8−𝑥 2
3√𝑥 2 + 𝑦 2
∫ ∫ d𝑦 d𝑥
0 𝑥 √8 𝜋
is equal to __________ (answer in integer)

MS 37/41

JAM 2024 Mathematical Statistics (MS)

Q.53 For some 𝑎 ≤ 0 and 𝑏 ∈ ℝ , let
0 𝑎 𝑏
1 1 1
−
𝐴= √2 √6 √3 .
1 1 1
( √2 √6 √3)
If 𝐴 is an orthogonal matrix, then the value of 𝑎√6 + 4𝑏√3 is equal to
__________ (answer in integer)

Q.54 Two factories 𝐹1 and 𝐹2 produce cricket bats that are labelled. Any randomly
chosen bat produced by factory 𝐹1 is defective with probability 0.5 and any
randomly chosen bat produced by factory 𝐹2 is defective with probability 0.1.
One of the factories is chosen at random, and two bats are randomly purchased
from the chosen factory. Let the labels on these purchased bats be 𝐵1 and 𝐵2 . If
𝐵1 is found to be defective, then the conditional probability that 𝐵2 is also
defective is equal to __________ (round off to 2 decimal places)

MS 38/41

JAM 2024 Mathematical Statistics (MS)

Q.55 Let 𝑋 be a discrete random variable with 𝑃(𝑋 ∈ {−5, −3, 0, 3, 5}) = 1.
Suppose that

𝑃(𝑋 = −3) = 𝑃(𝑋 = −5),

𝑃(𝑋 = 3) = 𝑃(𝑋 = 5) and

𝑃(𝑋 > 0) = 𝑃(𝑋 = 0) = 𝑃(𝑋 < 0).

Then the value of 12𝑃(𝑋 = 3) is equal to __________ (answer in integer)

Q.56 1
Consider a coin for which the probability of obtaining head in a single toss is 3.

Sunita tosses the coin once. If head appears, she receives a random amount of 𝑋
1
rupees, where 𝑋 has the 𝐸𝑥𝑝 (9) distribution. If tail appears, she loses a random
1
amount of 𝑌 rupees, where 𝑌 has the 𝐸𝑥𝑝 (3) distribution. Her expected gain

(in rupees) is equal to __________ (answer in integer)

MS 39/41

JAM 2024 Mathematical Statistics (MS)

Q.57 Let Θ be a random variable having 𝑈(0, 2𝜋) distribution. Let 𝑋 = cos Θ and
𝑌 = sin Θ. Let 𝜌 be the correlation coefficient between 𝑋 and 𝑌. Then 100𝜌 is
equal to __________ (answer in integer)

Q.58 Let 𝑋1 , 𝑋2 , … , 𝑋10 be a random sample from a 𝑈(−𝜃, 𝜃) distribution, where
𝜃 ∈ (0, ∞). Let 𝑋(10) = max{𝑋1 , 𝑋2 , … , 𝑋10 } and 𝑋(1) = min{𝑋1 , 𝑋2 , … , 𝑋10 }. If
the observed values of 𝑋(10) and 𝑋(1) are 8 and −10, respectively, then the
maximum likelihood estimate of 𝜃 is equal to __________ (answer in integer)

MS 40/41

JAM 2024 Mathematical Statistics (MS)

Q.59 Suppose that the weights (in kgs) of six months old babies, monitored at a
healthcare facility, have 𝑁(𝜇, 𝜎 2 ) distribution, where 𝜇 ∈ ℝ and 𝜎 > 0 are
unknown parameters. Let 𝑋1 , 𝑋2 , … , 𝑋9 be a random sample of the weights of
1 1
such babies. Let 𝑋̅ = 9 ∑9𝑖=1 𝑋𝑖 , 𝑆 = √8 ∑9𝑖=1(𝑋𝑖 − 𝑋̅)2 and let a

95% confidence interval for 𝜇 based on 𝑡-distribution be of the form

(𝑋̅ − ℎ(𝑆), 𝑋̅ + ℎ(𝑆)),
for an appropriate function ℎ of random variable 𝑆. If the observed values of 𝑋̅
and 𝑆 2 are 9 and 9.5, respectively, then the width of the confidence interval is
equal to __________ (round off to 2 decimal places)

(You may use 𝑡9,0.025 = 2.262, 𝑡8,0.025 = 2.306, 𝑡9,0.05 = 1.833, 𝑡8,0.05 = 1.86)

Q.60 Let 𝑋1 , 𝑋2 , 𝑋3 be a random sample from a Poisson distribution with mean
1
𝜆 , 𝜆 > 0. For testing 𝐻0 : 𝜆 = 8 against 𝐻1 : 𝜆 = 1, a test rejects 𝐻0 if and only if

𝑋1 + 𝑋2 + 𝑋3 > 1. The power of this test is equal to __________

(round off to 2 decimal places)

MS 41/41

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