JAM 2021 Question Paper Mathematical Statistics (MS)

JAM 2021 MATHEMATICAL STATISTICS - MS

Paper Specific Instructions

1. The examination is of 3 hours duration. There are a total of 60 questions carrying 100 marks. The entire
paper is divided into three sections, A, B and C. All sections are compulsory. Questions in each section are
of different types.

2. Section – A contains a total of 30 Multiple Choice Questions (MCQ). Each MCQ type question has four
choices out of which only one choice is the correct answer. Questions Q.1 – Q.30 belong to this section
and carry a total of 50 marks. Q.1 – Q.10 carry 1 mark each and Questions Q.11 – Q.30 carry 2 marks
each.

3. Section – B contains a total of 10 Multiple Select Questions (MSQ). Each MSQ type question is similar
to MCQ but with a difference that there may be one or more than one choice(s) that are correct out of the
four given choices. The candidate gets full credit if he/she selects all the correct answers only and no
wrong answers. Questions Q.31 – Q.40 belong to this section and carry 2 marks each with a total of 20
marks.

4. Section – C contains a total of 20 Numerical Answer Type (NAT) questions. For these NAT type
questions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor.
No choices will be shown for these type of questions. Questions Q.41 – Q.60 belong to this section and
carry a total of 30 marks. Q.41 – Q.50 carry 1 mark each and Questions Q.51 – Q.60 carry 2 marks each.

5. In all sections, questions not attempted will result in zero mark. In Section – A (MCQ), wrong answer will
result in NEGATIVE marks. For all 1 mark questions, 1/3 marks will be deducted for each wrong answer.
For all 2 marks questions, 2/3 marks will be deducted for each wrong answer. In Section – B (MSQ), there
is NO NEGATIVE and NO PARTIAL marking provisions. There is NO NEGATIVE marking in
Section – C (NAT) as well.

6. Only Virtual Scientific Calculator is allowed. Charts, graph sheets, tables, cellular phone or other
electronic gadgets are NOT allowed in the examination hall.

7. The Scribble Pad will be provided for rough work.

MS 1/21

JAM 2021 MATHEMATICAL STATISTICS - MS

Special Instructions/ Useful Data
ℝ The set of real numbers
ℝ𝑛 {(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) ∶ 𝑥𝑖 ∈ ℝ, 𝑖 = 1, 2, … , 𝑛}
det(𝑀) Determinant of a matrix 𝑀
𝐼𝑛 Identity matrix of order 𝑛 × 𝑛, 𝑛 = 2, 3, …
𝑔′ First derivative of a real valued function 𝑔
𝑔′′ Second derivative of a real valued function 𝑔
𝐹𝑐 Complement of an event 𝐹
𝑃(𝐹) Probability of an event 𝐹
𝑃(𝐹|𝐺) Conditional probability of an event 𝐹 given the occurrence of event 𝐺
𝑋∼𝑓 The probability density/mass function of the random variable 𝑋 is 𝑓
𝐸(𝑋) Expectation of a random variable 𝑋
Var(𝑋) Variance of a random variable 𝑋
𝑈(𝑎, 𝑏) Continuous uniform distribution on the interval (𝑎, 𝑏), −∞ < 𝑎 < 𝑏 < ∞
Poisson(𝜃) Poisson distribution with mean 𝜃, 𝜃 ∈ (0, ∞)
𝑁(𝜇, 𝜎 2 ) Normal distribution with mean 𝜇 and variance 𝜎 2 , 𝜇 ∈ (−∞, ∞), 𝜎 2 ∈ (0, ∞)
𝜒𝑛2 Central chi-square distribution with 𝑛 degrees of freedom, 𝑛 = 1,2, …
𝐹𝑚,𝑛 𝐹 distribution with (𝑚, 𝑛) degrees of freedom, 𝑚, 𝑛 = 1,2, …
Φ(⋅) Distribution function of 𝑁(0, 1)
|𝑥| Absolute value of 𝑥
MLE Maximum Likelihood Estimator
𝑛! 𝑛 ⋅ (𝑛 − 1) ⋯ 3 ⋅ 2 ⋅ 1, 𝑛 = 1, 2, 3, … , and 0! = 1
𝑛 𝑛! 0
( ) , 𝑘 = 0,1,2, … , 𝑛 and 𝑛 = 1, 2, … ; ( ) = 1
𝑘 𝑘! (𝑛 − 𝑘)! 0
max{𝑎1 , 𝑎2 , … , 𝑎𝑛 } Maximum of real numbers 𝑎1 , 𝑎2 , … , 𝑎𝑛 (𝑛 ≥ 2)
min{𝑎1 , 𝑎2 , … , 𝑎𝑛 } Minimum of real numbers 𝑎1 , 𝑎2 , … , 𝑎𝑛 (𝑛 ≥ 2)
ln 𝑥 Natural logarithm of 𝑥

MS 2/21

JAM 2021 MATHEMATICAL STATISTICS - MS

SECTION – A

MULTIPLE CHOICE QUESTIONS (MCQ)

Q. 1 – Q.10 carry one mark each.

Q.1 The value of the limit
1
1 2 𝑛 𝑛
lim ((1 + ) (1 + ) ⋯ (1 + ))
𝑛→∞ 𝑛 𝑛 𝑛
is equal to

1 3 4
(A) 𝑒 (B) (C) (D)
𝑒 𝑒 𝑒

Q.2 Let 𝑓: ℝ → ℝ be a function defined by
𝑓(𝑥) = 𝑥 7 + 5 𝑥 3 + 11 𝑥 + 15, 𝑥 ∈ ℝ.
Then, which of the following statements is TRUE?

(A) 𝑓 is both one-one and onto

(B) 𝑓 is neither one-one nor onto

(C) 𝑓 is one-one but NOT onto

(D) 𝑓 is onto but NOT one-one

Q.3 The value of the limit
𝑒 −3𝑥 − 𝑒 𝑥 + 4𝑥
lim
𝑥→0 5(1 − cos 𝑥 )

is equal to

(A) 1 (B) 0 2 8
(C) (D)
5 5

MS 3/21

JAM 2021 MATHEMATICAL STATISTICS - MS

Q.4 The value of the limit
𝑛
2𝑛 1
lim ∑ ( ) 𝑛
𝑛→∞ 𝑘 4
𝑘=0

is equal to

(A) 1 1 (C) 0 1
(B) (D)
2 4

Q.5 Let {𝑋𝑛 }𝑛≥1 be a sequence of independent and identically distributed random variables with
probability density function
1, if 0 < 𝑥 < 1
𝑓 (𝑥) = { .
0, otherwise
Then, the value of the limit
𝑛
1 1
lim 𝑃 (− ∑ ln 𝑋𝑖 ≤ 1 + )
𝑛→∞ 𝑛 √𝑛
𝑖=1

is equal to

1 (B) Φ(1) (C) 0 (D) Φ(2)
(A)
2

Q.6 Let 𝑋 be a 𝑈 (0, 1) random variable and let 𝑌 = 𝑋 2 . If 𝜌 is the correlation coefficient between the
random variables 𝑋 and 𝑌, then 48 𝜌2 is equal to

(A) 48 (B) 45 (C) 35 (D) 30

Q.7 1 1 0
Let 𝑀 be a 3 × 3 real matrix. Let (2) , (1) and (−1) be the eigenvectors of 𝑀 corresponding to
3 1 𝛼
three distinct eigenvalues of 𝑀, where 𝛼 is a real number. Then, which of the following is NOT a
possible value of 𝛼?

(A) 0 (B) 1 (C) −2 (D) 2

MS 4/21

JAM 2021 MATHEMATICAL STATISTICS - MS

Q.8 If the series ∑∞
𝑛=1 𝑎𝑛 converges absolutely, then which of the following series diverges?

∑∞ 𝑎 +𝑎𝑛+1
𝑛=1 |𝑎2𝑛 | ∑∞
(A) 𝑛
(B) 𝑛=1 2

(C) ∑∞
𝑛=1(𝑎𝑛 )
3
∑∞
1
(D) 𝑛=2 ((ln 𝑛)2
+ 𝑎𝑛 )

Q.9 There are three urns, labeled, Urn 1, Urn 2 and Urn 3. Urn 1 contains 2 white balls and 2 black
balls, Urn 2 contains 1 white ball and 3 black balls and Urn 3 contains 3 white balls and 1 black
ball. Consider two coins with probability of obtaining head in their single trials as 0.2 and 0.3. The
two coins are tossed independently once, and an urn is selected according to the following scheme:
Urn 1 is selected if 2 heads are obtained; Urn 3 is selected if 2 tails are obtained; otherwise Urn 2 is
selected. A ball is then drawn at random from the selected urn. Then
𝑃(Urn 1 is selected | the ball drawn is white )
is equal to

6 12 1 1
(A) (B) (C) (D)
109 109 18 9

Q.10 Let 𝑋 be a random variable with probability density function
1 −|𝑥|
𝑓 (𝑥) = 𝑒 , − ∞ < 𝑥 < ∞.
2
Then, which of the following statements is FALSE?

(A) 𝐸 (𝑋 |𝑋|) = 0

(B) 𝐸 (𝑋 |𝑋|2 ) = 0

𝑋
(C) 𝐸 (|𝑋| sin ( )) = 0
|𝑋|

𝑋
(D) 𝐸 (|𝑋| sin2 ( )) = 0
|𝑋|

MS 5/21

JAM 2021 MATHEMATICAL STATISTICS - MS

Q. 11 – Q. 30 carry two marks each.

Q.11 Let 𝑓: ℝ2 → ℝ be a function defined by
𝑦3
, (𝑥, 𝑦) ≠ (0,0)
𝑓(𝑥, 𝑦) = {𝑥 2 + 𝑦 2 .
0, ( )
𝑥, 𝑦 = (0,0)
Let 𝑓𝑥 (𝑥, 𝑦) and 𝑓𝑦 (𝑥, 𝑦) denote the first order partial derivatives of 𝑓(𝑥, 𝑦) with respect to 𝑥 and 𝑦,
respectively, at the point (𝑥, 𝑦). Then, which of the following statements is FALSE?

(A) 𝑓𝑥 (𝑥, 𝑦) exists and is bounded at every (𝑥, 𝑦) ∈ ℝ2

(B) 𝑓𝑦 (𝑥, 𝑦) exists and is bounded at every (𝑥, 𝑦) ∈ ℝ2

(C) 𝑓𝑦 (0,0) exists and 𝑓𝑦 (𝑥, 𝑦) is continuous at (0,0)

(D) 𝑓 is NOT differentiable at (0,0)

Q.12 Let {𝑋𝑛 }𝑛≥1 be a sequence of independent and identically distributed 𝑁(0, 1) random variables.
Then,
∑𝑛𝑖=1 𝑋𝑖4 − 3𝑛
lim 𝑃 ( ≤ √6)
𝑛→∞ √32 𝑛
is equal to

1 (B) Φ(√2 ) (C) 0 (D) Φ(1)
(A)
2

Q.13 1
Consider a sequence of independent Bernoulli trials with probability of success in each trial as .
3
The probability that three successes occur before four failures is equal to

179 179 233 179
(A) (B) (C) (D)
243 841 729 1215

MS 6/21

JAM 2021 MATHEMATICAL STATISTICS - MS

Q.14 |𝑋|
Let 𝑋 and 𝑌 be independent 𝑁(0, 1) random variables and 𝑍 = . Then, which of the
|𝑌|

following expectations is finite?

1 (C) 𝐸(𝑍) 1
(A) 𝐸( ) (B) 𝐸(𝑍√𝑍) (D) 𝐸 (𝑍√𝑍 )
√𝑍

Q.15 1 1 3
Consider three coins having probabilities of obtaining head in a single trial as , and ,
4 2 4
respectively. A player selects one of these three coins at random (each coin is equally likely to be
selected). If the player tosses the selected coin five times independently, then the probability of
obtaining two tails in five tosses is equal to

85 255
(A) (B)
384 384

125 64
(C) (D)
384 384

Q.16 Let 𝑋 be a random variable having the probability density function
𝑒 −𝑥 , 𝑥 > 0
𝑓 (𝑥) = { .
0, 𝑥 ≤ 0
Define 𝑌 = [𝑋], where [𝑋] denotes the largest integer not exceeding 𝑋. Then, 𝐸(𝑌 2 ) is equal to

𝑒(𝑒+1) 𝑒+1
(A) (B)
𝑒−1 (𝑒−1)2

𝑒(𝑒+1)2 (𝑒+1)2
(C) (D)
𝑒−1 (𝑒−1)2

MS 7/21

JAM 2021 MATHEMATICAL STATISTICS - MS

Q.17 Let 𝑋 be a continuous random variable having the moment generating function
𝑒𝑡 − 1
, 𝑀(𝑡) =
𝑡 ≠ 0.
𝑡
Let 𝛼 = 𝑃(48 𝑋 2 − 40 𝑋 + 3 > 0) and 𝛽 = 𝑃((ln 𝑋)2 + 2 ln 𝑋 − 3 > 0).
Then, the value of 𝛼 − 2 ln 𝛽 is equal to

10 19 13 17
(A) (B) (C) (D)
3 3 3 3

Q.18 Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 (𝑛 ≥ 3) be a random sample from Poisson(𝜃), where 𝜃 ∈ (0, ∞) is unknown and
let
𝑛

𝑇 = ∑ 𝑋𝑖 .
𝑖=1

Then, the uniformly minimum variance unbiased estimator of 𝑒 −2𝜃 𝜃 3

𝑇 𝑇 𝑇 2 𝑇−3
(A) is ( − 1) ( − 2) (1 − )
𝑛 𝑛 𝑛 𝑛

𝑇(𝑇−1)(𝑇−2)(𝑛−2)𝑇−3
(B) is
𝑛𝑇

(C) does NOT exist

−
2𝑇
𝑇 3
(D) is 𝑒 𝑛 (𝑛 )

Q.19 Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 (𝑛 ≥ 2) be a random sample from 𝑈(𝜃 − 5, 𝜃 + 5), where 𝜃 ∈ (0, ∞) is
unknown. Let 𝑇 = max{𝑋1 , 𝑋2 , … , 𝑋𝑛 } and 𝑈 = min{𝑋1 , 𝑋2 , … , 𝑋𝑛 }. Then, which of the following
statements is TRUE?

𝑇+𝑈
(A) is the unique MLE of 𝜃
2

2 1
(B) is an MLE of 𝜃
𝑇+𝑈

1
(C) MLE of 𝜃 does NOT exist

(D) 𝑈 + 8 is an MLE of 𝜃

MS 8/21

JAM 2021 MATHEMATICAL STATISTICS - MS

Q.20 Let 𝑋 and 𝑌 be random variables having chi-square distributions with 6 and 3 degrees of freedom,
respectively. Then, which of the following statements is TRUE?

(A) 𝑃 (𝑋 > 0.7) > 𝑃(𝑌 > 0.7)

(B) 𝑃 (𝑋 > 0.7) < 𝑃(𝑌 > 0.7)

(C) 𝑃 (𝑋 > 3) < 𝑃(𝑌 > 3)

(D) 𝑃 (𝑋 < 6) > 𝑃(𝑌 < 6)

Q.21 Let (𝑋, 𝑌) be a random vector with joint moment generating function
1
𝑀(𝑡1 , 𝑡2 ) = , −∞ < 𝑡1 < ∞, −∞ < 𝑡2 < min{1, 1 − 𝑡1 }.
(1 − 𝑡1 + 𝑡2 ))(1 − 𝑡2 )
(
Let 𝑍 = 𝑋 + 𝑌. Then, Var(𝑍) is equal to

(A) 3 (B) 4 (C) 5 (D) 6

Q.22 Let 𝑋 be a continuous random variable with distribution function
0, if 𝑥 < 0
𝐹 (𝑥) = {𝑎 𝑥 2 , if 0 ≤ 𝑥 < 2,
1, if 𝑥 ≥ 2
for some real constant 𝑎. Then, 𝐸(𝑋) is equal to

4 1 (C) 1 (D) 0
(A) (B)
3 4

MS 9/21

JAM 2021 MATHEMATICAL STATISTICS - MS

Q.23 Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 be a random sample from an exponential distribution with probability density
function
𝜃𝑒 −𝜃𝑥 , 𝑥>0
𝑓 (𝑥; 𝜃) = { ,
0, otherwise
where 𝜃 ∈ (0, ∞) is unknown. Let 𝛼 ∈ (0,1) be fixed and let 𝛽 be the power of the most powerful
test of size 𝛼 for testing 𝐻0 : 𝜃 = 1 against 𝐻1 : 𝜃 = 2.
Consider the critical region
𝑛
1 2
𝑅 = {(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) ∈ ℝ𝑛 ∶ ∑ 𝑥𝑖 > 𝜒 (1 − 𝛼 )} ,
2 2𝑛
𝑖=1

2 ( ) 2 2 ( ))
where for any 𝛾 ∈ (0, 1), 𝜒2𝑛 𝛾 is a fixed point such that 𝑃 (𝜒2𝑛 > 𝜒2𝑛 𝛾 = 𝛾. Then, the

critical region 𝑅 corresponds to the

(A) most powerful test of size 𝛼 for testing 𝐻0 : 𝜃 = 1 against 𝐻1 : 𝜃 = 2

(B) most powerful test of size 1 − 𝛼 for testing 𝐻0∗ : 𝜃 = 2 against 𝐻1∗ : 𝜃 = 1

(C) most powerful test of size 𝛽 for testing 𝐻0∗ : 𝜃 = 2 against 𝐻1∗ : 𝜃 = 1

(D) most powerful test of size 1 − 𝛽 for testing 𝐻0∗ : 𝜃 = 2 against 𝐻1∗ : 𝜃 = 1

Q.24 Let
∞ ∞
1 1 𝑘 1 1 𝑘
𝑆= ∑(−1)𝑘−1 ( ) and 𝑇 = ∑ ( ) .
𝑘 4 𝑘 5
𝑘=1 𝑘=1

Then, which of the following statements is TRUE?

(A) 𝑆 − 𝑇 = 0 (B) 5𝑆−4𝑇 = 0

(C) 4 𝑆 − 5 𝑇 = 0 (D) 16 𝑆 − 25 𝑇 = 0

Q.25 Let 𝐸1 , 𝐸2 , 𝐸3 and 𝐸4 be four events such that
2 1 1
𝑃(𝐸𝑖 |𝐸4 ) = , 𝑖 = 1, 2, 3; 𝑃(𝐸𝑖 ∩ 𝐸𝑗𝑐 |𝐸4 ) = , 𝑖, 𝑗 = 1,2,3; 𝑖 ≠ 𝑗 and 𝑃(𝐸1 ∩ 𝐸2 ∩ 𝐸3𝑐 |𝐸4 ) =
3 6 6
Then, 𝑃 (𝐸1 ∪ 𝐸2 ∪ 𝐸3 |𝐸4 ) is equal to

1 2 5 7
(A) (B) (C) (D)
2 3 6 12

MS 10/21

JAM 2021 MATHEMATICAL STATISTICS - MS

Q.26 Let 𝑎1 = 5 and define recursively
1 3
𝑎𝑛+1 = 34 (𝑎𝑛 )4 , 𝑛 ≥ 1.
Then, which of the following statements is TRUE?

(A) {𝑎𝑛 } is monotone increasing, and lim 𝑎𝑛 = 3
𝑛→∞

(B) {𝑎𝑛 } is monotone decreasing, and lim 𝑎𝑛 = 3
𝑛→∞

(C) {𝑎𝑛 } is non-monotone, and lim 𝑎𝑛 = 3
𝑛→∞

(D) {𝑎𝑛 } is decreasing, and lim 𝑎𝑛 = 0
𝑛→∞

Q.27 Consider the problem of testing 𝐻0 : 𝑋 ∼ 𝑓0 against 𝐻1 : 𝑋 ∼ 𝑓1 based on a sample of size 1, where
1, 0 ≤ 𝑥 ≤ 1 2 − 2𝑥, 0 ≤ 𝑥 ≤ 1
𝑓0 (𝑥) = { and 𝑓1 (𝑥) = {
0, otherwise 0, otherwise
Then, the probability of Type II error of the most powerful test of size 𝛼 = 0.1 is equal to

(A) 0.81 (B) 0.91 (C) 0.1 (D) 1

Q.28 For 𝑎 ∈ ℝ, consider the system of linear equations
𝑎𝑥+𝑎𝑦 = 𝑎+2
𝑥 + 𝑎 𝑦 + (𝑎 − 1)𝑧 = 𝑎−4
𝑎 𝑥 + 𝑎 𝑦 + (𝑎 − 2)𝑧 = −8
in the unknowns 𝑥, 𝑦 and 𝑧. Then, which of the following statements is TRUE?

(A) The given system has a unique solution for 𝑎 = 1

(B) The given system has infinitely many solutions for 𝑎 = 2

(C) The given system has a unique solution for 𝑎 = −2

(D) The given system has infinitely many solutions for 𝑎 = −2

MS 11/21

JAM 2021 MATHEMATICAL STATISTICS - MS

Q.29 Let {𝑎𝑛 }𝑛 ≥ 1 be a sequence of real numbers such that 𝑎𝑛 ≥ 1, for all 𝑛 ≥ 1. Then, which of the
following conditions imply the divergence of {𝑎𝑛 }𝑛 ≥ 1 ?

(A) {𝑎𝑛 }𝑛 ≥ 1 is non-increasing

(B) ∑∞
𝑛=1 𝑏𝑛 converges, where 𝑏1 = 𝑎1 and 𝑏𝑛 = 𝑎𝑛+1 − 𝑎𝑛 , for all 𝑛 > 1

𝑎2𝑛+1 1
(C) lim =2
𝑛→∞ 𝑎2𝑛

(D) {√𝑎𝑛 } converges
𝑛≥1

Q.30 4 1 9
Let 𝐸1 , 𝐸2 and 𝐸3 be three events such that 𝑃 (𝐸1 ) = 5 , 𝑃(𝐸2 ) = 2 and 𝑃(𝐸3 ) = 10 .

Then, which of the following statements is FALSE?

9 1
(A) 𝑃 (𝐸1 ∪ 𝐸2 ∪ 𝐸3 ) ≥ 10 (B) 𝑃(𝐸2 ∩ 𝐸3 ) ≤ 2

1 4
(C) 𝑃(𝐸1 ∩ 𝐸2 ∩ 𝐸3 ) ≤ (D) 𝑃(𝐸1 ∪ 𝐸2 ) ≥
6 5

MS 12/21

JAM 2021 MATHEMATICAL STATISTICS - MS

SECTION - B

MULTIPLE SELECT QUESTIONS (MSQ)

Q. 31 – Q. 40 carry two marks each.

Q.31 Consider the linear system 𝐴 𝑥 = 𝑏, where 𝐴 is an 𝑚 × 𝑛 matrix, 𝑥 is an 𝑛 × 1 vector of unknowns
and 𝑏 is an 𝑚 × 1 vector. Further, suppose there exists an 𝑚 × 1 vector 𝑐 such that the linear
system 𝐴𝑥 = 𝑐 has NO solution. Then, which of the following statements is/are necessarily
TRUE?

(A) If 𝑚 ≤ 𝑛 and 𝑑 is the first column of 𝐴, then the linear system 𝐴𝑥 = 𝑑 has a unique solution

(B) If 𝑚 ≥ 𝑛, then Rank(𝐴) < 𝑛

(C) Rank(𝐴) < 𝑚

(D) If 𝑚 > 𝑛, then the linear system 𝐴𝑥 = 0 has a solution other than 𝑥 = 0

Q.32 Let 𝐴 be a 3 × 3 real matrix such that 𝐴 ≠ 𝐼3 and the sum of the entries in each row of 𝐴 is 1.
Then, which of the following statements is/are necessarily TRUE?

(A) 𝐴 − 𝐼3 is an invertible matrix

(B) The set { 𝑥 ∈ ℝ3 : (𝐴 − 𝐼3 )𝑥 = 0 } has at least two elements (𝑥 is a column vector)

(C) The characteristic polynomial, 𝑝(𝜆), of 𝐴 + 2 𝐴2 + 𝐴3 has (𝜆 − 4) as a factor

(D) 𝐴 cannot be an orthogonal matrix

MS 13/21

JAM 2021 MATHEMATICAL STATISTICS - MS

Q.33 Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 be a random sample from 𝑁(𝜃, 1), where 𝜃 ∈ (−∞, ∞) is unknown. Consider the
problem of testing 𝐻0 : 𝜃 ≤ 0 against 𝐻1 : 𝜃 > 0. Let 𝛽 (𝜃) denote the power function of the
likelihood ratio test of size 𝛼 (0 < 𝛼 < 1) for testing 𝐻0 against 𝐻1 . Then, which of the following
statements is/are TRUE?

(A) 𝛽 (𝜃) > 𝛽(0), for all 𝜃 > 0

(B) 𝛽 (𝜃) < 𝛽(0), for all 𝜃 > 0

(C) The critical region of the likelihood test of size 𝛼 is
∑𝑛𝑖=1 𝑥𝑖
{(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) ∈ ℝ𝑛 : √𝑛 > 𝜏𝛼⁄2 } ,
𝑛
𝛼
where 𝜏𝛼⁄2 is a fixed point such that 𝑃(𝑍 > 𝜏𝛼⁄2 ) = 2 , 𝑍 ∼ 𝑁(0, 1).

(D) The critical region of the likelihood test of size 𝛼 is
∑𝑛𝑖=1 𝑥𝑖
{(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) ∈ ℝ𝑛 : √𝑛 < 𝜏𝛼 } ,
𝑛
where 𝜏𝛼 is a fixed point such that 𝑃(𝑍 > 𝜏𝛼 ) = 𝛼, 𝑍 ∼ 𝑁(0, 1).

Q.34 Consider the function
𝑓(𝑥, 𝑦) = 3 𝑥 2 + 4 𝑥 𝑦 + 𝑦 2 , (𝑥, 𝑦) ∈ ℝ2 .
If 𝑆 = {(𝑥, 𝑦) ∈ ℝ2 ∶ 𝑥 2 + 𝑦 2 = 1}, then which of the following statements is/are TRUE?

(A) The maximum value of 𝑓 on 𝑆 is 3 + √5

(B) The minimum value of 𝑓 on 𝑆 is 3 − √5

(C) The maximum value of 𝑓 on 𝑆 is 2 + √5

(D) The minimum value of 𝑓 on 𝑆 is 2 − √5

MS 14/21

JAM 2021 MATHEMATICAL STATISTICS - MS

Q.35 Let 𝑓: ℝ → ℝ be a twice differentiable function. Then, which of the following statements is/are
necessarily TRUE?

(A) 𝑓′′ is continuous

(B) If 𝑓 ′ (0) = 𝑓 ′ (1), then 𝑓 ′′ (𝑥) = 0 has a solution in (0, 1)

(C) 𝑓′ is bounded on [8, 10]

(D) 𝑓 ′′ is bounded on (0, 1)

Q.36 Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 (𝑛 ≥ 2) be independent and identically distributed random variables with
probability density function
1
𝑓 (𝑥) = { 𝑥 2 , 𝑥≥1 .
0, otherwise
Then, which of the following random variables has/have finite expectation?

(A) 𝑋1 1 (C) √𝑋1 (D) min{𝑋1 , … , 𝑋𝑛 }
(B)
𝑋2

Q.37 A sample of size 𝑛 is drawn randomly (without replacement) from an urn containing 5𝑛2 balls, of
which 2𝑛2 are red balls and 3𝑛2 are black balls. Let 𝑋𝑛 denote the number of red balls in the
𝐸(𝑋𝑛) Var(𝑋𝑛)
selected sample. If ℓ = lim 𝑛
and 𝑚 = lim 𝑛
, then which of the following statements
𝑛→∞ 𝑛→∞

is/are TRUE?

16 3
(A) ℓ + 𝑚 = 25 (B) ℓ − 𝑚 = 25

14 ℓ 5
(C) ℓ𝑚 = 125 (D) =
𝑚 3

MS 15/21

JAM 2021 MATHEMATICAL STATISTICS - MS

Q.38 Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 (𝑛 ≥ 2) be a random sample from a distribution with probability density function
1
( ) , −𝜃 ≤ 𝑥 ≤ 𝜃
𝑓 𝑥; 𝜃 = {2𝜃 ,
0, |𝑥| > 𝜃
where 𝜃 ∈ (0, ∞) is unknown. If 𝑅 = min{𝑋1 , 𝑋2 , … , 𝑋𝑛 } and 𝑆 = max{𝑋1 , 𝑋2 , … , 𝑋𝑛 }, then which
of the following statements is/are TRUE?

(A) (𝑅, 𝑆) is jointly sufficient for 𝜃

(B) 𝑆 is an MLE of 𝜃

(C) max{|𝑋1 |, |𝑋2 |, … , |𝑋𝑛 |} is a complete and sufficient statistic for 𝜃

𝑅
(D) Distribution of does NOT depend on 𝜃
𝑆

Q.39 Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 (𝑛 ≥ 2) be a random sample from a distribution with probability density function
3𝑥 2 −𝑥 3/𝜃
𝑓 (𝑥; 𝜃) = { 𝜃 𝑒 , 𝑥>0 ,
0, otherwise
where 𝜃 ∈ (0, ∞) is unknown.
If 𝑇 = ∑𝑛𝑖=1 𝑋𝑖3 , then which of the following statements is/are TRUE?

𝑛−1 1
(A) is the unique uniformly minimum variance unbiased estimator of
𝑇 𝜃

𝑛 1
(B) is the unique uniformly minimum variance unbiased estimator of
𝑇 𝜃

1 1
(C) (𝑛 − 1) ∑𝑛
𝑖=1 3 is the unique uniformly minimum variance unbiased estimator of 𝜃
𝑋𝑖

𝑛 1
(D) is the MLE of
𝑇 𝜃

MS 16/21

JAM 2021 MATHEMATICAL STATISTICS - MS

Q.40 Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 (𝑛 ≥ 2) be a random sample from a distribution with probability density function
𝜃𝑥 𝜃−1 , 0≤𝑥≤1
𝑓(𝑥; 𝜃) = { ,
0, otherwise
where 𝜃 ∈ (0, ∞) is unknown. Then, which of the following statements is/are TRUE?

𝜃6
(A) Cramer-Rao lower bound, based on 𝑋1 , 𝑋2 , … , 𝑋𝑛 , for the estimand 𝜃 3 is 9
𝑛

𝜃2
(B) Cramer-Rao lower bound, based on 𝑋1 , 𝑋2 , … , 𝑋𝑛 , for the estimand 𝜃 3 is
𝑛

1
(C) There does NOT exist any unbiased estimator of which attains the Cramer-Rao lower
𝜃
bound

1
(D) There exists an unbiased estimator of which attains the Cramer-Rao lower bound
𝜃

MS 17/21

JAM 2021 MATHEMATICAL STATISTICS - MS

SECTION – C

NUMERICAL ANSWER TYPE (NAT)

Q. 41 – Q. 50 carry one mark each.

Q.41 0 1 0
Let 𝛼, 𝛽 and 𝛾 be the eigenvalues of 𝑀 = [ 1 3 3]. If 𝛾 = 1 and 𝛼 > 𝛽, then the value of
−1 2 2
2 𝛼 + 3 𝛽 is _______________________.

Q.42 5 −6
Let 𝑀 = ( ) be a 2 × 2 matrix. If 𝛼 = det(𝑀4 − 6𝐼2 ), then the value of 𝛼 2 is
3 −4
___________________.

Q.43 Let 𝑆 = {(𝑥, 𝑦) ∈ ℝ2 ∶ 2 ≤ 𝑥 ≤ 𝑦 ≤ 4}. Then, the value of the integral

1
∬ 𝑑𝑥 𝑑𝑦
4−𝑥
𝑆

is _________________________.

Q.44 Let 𝐴 = {(𝑥, 𝑦, 𝑧) ∈ ℝ3 : 0 ≤ 𝑥 ≤ 𝑦 ≤ 𝑧 ≤ 1}. Let 𝛼 be the value of the integral

∭ 𝑥 𝑦 𝑧 𝑑𝑥 𝑑𝑦 𝑑𝑧.
𝐴
Then, 384 𝛼 is equal to ____________________________.

Q.45 Let 𝑓0 and 𝑓1 be the probability mass functions given by
𝑥 1 2 3 4 5 6
𝑓0 (𝑥) 0.1 0.1 0.1 0.1 0.1 0.5
𝑓1 (𝑥) 0.1 0.1 0.2 0.2 0.2 0.2

Consider the problem of testing the null hypothesis 𝐻0 : 𝑋 ∼ 𝑓0 against 𝐻1 : 𝑋 ∼ 𝑓1 based on a single
sample 𝑋. If 𝛼 and 𝛽, respectively, denote the size and power of the test with critical region
{𝑥 ∈ ℝ ∶ 𝑥 > 3}, then 10(𝛼 + 𝛽) is equal to _________________.

MS 18/21

JAM 2021 MATHEMATICAL STATISTICS - MS

Q.46 Let 5, 10, 4, 15, 6 be an observed random sample of size 5 from a distribution with probability
density function
𝑒 −(𝑥−𝜃) , 𝑥≥𝜃
𝑓 (𝑥; 𝜃) = {
0, otherwise
𝜃 ∈ (−∞, 3] is unknown. Then, the maximum likelihood estimate of 𝜃 based on the observed
sample is equal to ____________________.

Q.47 Let
2𝑛2
1
𝛼 = lim ∑
𝑛→∞
𝑚=𝑛2
√5 𝑛4 + 𝑛3 + 𝑚

Then, 10 √5 𝛼 is equal to ____________________________.

Q.48 Let 𝑋 be a random variable having the probability density function
1 𝑥2 𝑥2
𝑓 (𝑥) = (2 𝑒 − 2 + 3 𝑒 − 8 ) , −∞ < 𝑥 < ∞.
8√2𝜋
Then, 4 𝐸(𝑋 4 ) is equal to __________________.

Q.49 Let 𝑋 be a random variable with moment generating function
1 1 𝑡 1 2𝑡 1 −𝑡 1 −2𝑡
𝑀𝑋 (𝑡) = + 𝑒 + 𝑒 + 𝑒 + 𝑒 , 𝑡 ∈ ℝ.
12 6 3 4 6
Then, 8 𝐸(𝑋) is equal to ________________________.

𝜋
Q.50 Let 𝛽 denote the length of the curve 𝑦 = ln(sec 𝑥 ) from 𝑥 = 0 to 𝑥 = 4 .

Then, the value of 3 √2 (𝑒 𝛽 − 1) is equal to ______________.

MS 19/21

JAM 2021 MATHEMATICAL STATISTICS - MS

Q.51 – Q. 60 carry two marks each.

Q.51 Let 𝑆 ⊆ ℝ2 be the region bounded by the parallelogram with vertices at the points (1, 0), (3, 2),

(3, 5) and (1, 3). Then, the value of the integral ∬𝑆 (𝑥 + 2 𝑦) 𝑑𝑥 𝑑𝑦 is equal to _____________.

Q.52 1 1
Let 𝐴 = { (𝑥, 𝑦) ∈ ℝ2 : 𝑥 2 − 2√𝜋 < 𝑦 < 𝑥 2 + 2√𝜋 } and let the joint probability density function

of (𝑋, 𝑌) be
−(𝑥−1)2 (𝑥, 𝑦) ∈ 𝐴
𝑓(𝑥, 𝑦) = {𝑒 ,
0, otherwise
Then, the covariance between the random variables 𝑋 and 𝑌 is equal to ________________.

Q.53 Let 𝑋1 and 𝑋2 be independent 𝑁(0, 1) random variables. Define
−1, if 𝑢 < 0
( )
sgn 𝑢 = { 0, if 𝑢 = 0
1, if 𝑢 > 0
Let 𝑌1 = 𝑋1 sgn(𝑋2 ) and 𝑌2 = 𝑋2 sgn(𝑋1 ). If the correlation coefficient between 𝑌1 and 𝑌2 is 𝛼,
then 𝜋𝛼 is equal to __________________________.

Q.54 Let
𝑛
𝑛 2𝑘 (𝑛 − 2)𝑛−𝑘
𝑎𝑛 = ∑ ( ) , 𝑛 = 2, 3, ….
𝑘 𝑛𝑛
𝑘=2
2
Then, 𝑒 lim (1 − 𝑎𝑛 ) is equal to _______________________.
𝑛→∞

Q.55 1 1 1
Let 𝐸1 , 𝐸2 , 𝐸3 and 𝐸4 be four independent events such that 𝑃 (𝐸1 ) = 2 , 𝑃(𝐸2 ) = 3 , 𝑃 (𝐸3 ) = 4 and
1
𝑃 (𝐸4 ) = 5 . Let 𝑝 be the probability that at most two events among 𝐸1 , 𝐸2 , 𝐸3 and 𝐸4 occur.
Then, 240 𝑝 is equal to _____________.

Q.56 Let the random vector (𝑋, 𝑌) have the joint probability mass function
𝑥−𝑦+5
10 5 1 3 𝑦−𝑥+10
𝑓 (𝑥, 𝑦) = {( 𝑥 ) (𝑦) (4) ( )
4
, 𝑥 = 0,1, … , 10; 𝑦 = 0,1, … , 5
0, otherwise
Let 𝑍 = 𝑌 − 𝑋 + 10. If 𝛼 = 𝐸(𝑍) and 𝛽 = Var(𝑍), then 8 𝛼 + 48 𝛽 is equal to ___________.

MS 20/21

JAM 2021 MATHEMATICAL STATISTICS - MS

Q.57 Let 𝑆 = {(𝑥, 𝑦) ∈ ℝ2 : 0 ≤ 𝑥 ≤ 𝜋, min{sin 𝑥, cos 𝑥} ≤ 𝑦 ≤ max{sin 𝑥, cos 𝑥}}.

If 𝛼 is the area of 𝑆, then the value of 2√2 𝛼 is equal to __________________.

Q.58 The number of real roots of the polynomial
𝑓 (𝑥) = 𝑥11 − 13 𝑥 + 5
is ________________.

Q.59 3 2𝑛
Let 𝛼 = lim (1 + 𝑛 sin 2 ) . Then, ln 𝛼 is equal to ___________________.
𝑛→∞ 𝑛

Q.60 Let 𝜙: (−1, 1) → ℝ be defined by
𝑥4
1
𝜙(𝑥) = ∫ 𝑑𝑡.
7
1 + 𝑡3
𝑥
𝜙(𝑥)
If 𝛼 = lim 4 , then 42 𝛼 is equal to ___________________.
𝑥→0 𝑒 2 𝑥 −1

END OF THE QUESTION PAPER

MS 21/21