JAM 2026 Question Paper Mathematical Statistics (MS)

JAM 2026 Mathematical Statistics (MS)

Special Instructions / Useful Data

ℕ
Special Instructions / Useful Data

ℝ
The set of positive integers

ℝ , ,…, : ∈ ℝ, = 1, 2, … , , = 2, 3, …
The set of real numbers

× identity matrix, = 2, 3, 4, …
Derivative of the function

Transpose of the matrix
Complement of a set

|! given the occurrence of the event !
Probability of an event

" Expectation of a random variable "
Conditional probability of an event

#$% " Variance of a random variable "
&'( ", ) Covariance between the random variables " and )
* $, + Continuous uniform distribution on the interval $, + , −∞ < $ < + < ∞
/ 0 Exponential distribution with the probability density function, for 0 > 0,
1 67
= 30 5 , >0
8

0, otherwise
A B, C Normal distribution with mean B and variance C , B ∈ ℝ, C > 0
D
E Central Student’s E distribution with
Central chi-square distribution with degrees of freedom

D ,F A constant such that G" > D ,F H = I, where "~D and I ∈ 0,1
degrees of freedom

Φ ⋅ The cumulative distribution function of the A 0, 1 random variable
"M The %-th order statistic of a random sample " , " , … , " of size ,
% = 1, 2, … , ≥1

MS Page 1 of 47

JAM 2026 Mathematical Statistics (MS)

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

Q.1 If O denotes the radius of convergence of the power series
W
2 ⋅ 4 ⋅6⋯2
PQ V ,
2 ⋅ 5 ⋅ 8⋯ 3 − 1
X

then which one of the following statements is true?

(A) 9O = 4

(B) 4O = 9

(C) O = 1

(D) 2O = 1

MS Page 2 of 47

JAM 2026 Mathematical Statistics (MS)

Let Z: ℝ → ℝ be a function defined by Z = \a E − 9E + 8 ^E .
Q.2 _`

Which one of the following statements is NOT true?

(A) Z has a local minimum at =1

(B) Z has a point of inflection at =0

(C) = 0, = 1, = 2 are critical points of Z

(D) Z = 0 has at least two distinct solutions, where Z denotes the second
order derivative of Z

Q.3 Let be a 2 × 2 real symmetric matrix such that the trace of is 6 and the
determinant of is 5. Which one of the following statements is true?

(A) − is positive definite

(B) −3 is negative definite

(C) + −3 is positive definite

(D) −6 is negative definite

MS Page 3 of 47

JAM 2026 Mathematical Statistics (MS)

Q.4 Let " be a positive continuous random variable. Consider the transformation
) = " b . Then, the Jacobian of the inverse transformation is

d 6 è
(A) c
b

d è
(B)

d 6 è
(C)
b

d
f
e
(D)

MS Page 4 of 47

JAM 2026 Mathematical Statistics (MS)

Q.5 A retail shop sells three brands of tea, namely Brand A, Brand B, and Brand C,
and each in two varieties, namely regular and premium. The proportion of
customers buying the Brands A, B, and C are 40%, 35%, and 25%, respectively.
Out of those customers who buy Brand A, 30% buy the premium variety, out of
those who buy Brand B, 40% buy the premium variety, while out of those who
buy Brand C, 60% buy the premium variety. Given that a randomly selected
customer has bought the premium variety of tea, the probability that he/she has
bought Brand B equals

b
b
(A)

c
c
(B)

g
g
(C)

h
h
(D)

MS Page 5 of 47

JAM 2026 Mathematical Statistics (MS)

Q.6 Let ", ) be a random vector with #$% " = #$% ) = 1 and
&'( ", ) = −0.5. For which one of the following values of +, the random
variables * = +" + ) and # = " + +) are uncorrelated?

(A) 2 − √3

(B) √3 − 2

(C) √3

(D) 2

Q.7 Let " , " , … , " be a random sample of size ≥ 2 from a A 0, C
distribution, where C > 0 is an unknown parameter. Which one of the following
statements is true?

(A) ∑ X " is a sufficient statistic for C

(B) ∑ X " is a sufficient statistic for C

(C) ∑ X " is a complete statistic

(D) ∑ X " is a biased estimator of C

MS Page 6 of 47

JAM 2026 Mathematical Statistics (MS)

Q.8 Let " , " , … , " be a random sample of size ≥2 from a A B, 1
distribution, where B ∈ ℝ is an unknown parameter. Consider the problem of
testing la : B = 5 against l : B ≠ 5. Which one of the following statements is
true?

(A) The power function of the likelihood ratio test at level 0.05 has a local
maximum at B = 5

(B) The power function of the likelihood ratio test at level 0.05 is decreasing
on 5, ∞

(C) The power function of the likelihood ratio test at level 0.05 is decreasing

on −∞, 5

(D) The power function of the likelihood ratio test at level 0.05 is increasing

on 0, 5

MS Page 7 of 47

JAM 2026 Mathematical Statistics (MS)

Q.9 Consider a discrete time Markov chain with state space n = 1, 2 and the
transition probability diagram

If o = o , o is the stationary distribution of the Markov chain, then which
one of the following statements is true?

(A) o ,o =p , q

o ,o =p , q
c
g g
(B)

o ,o =p , q
c
b b
(C)

o ,o =p , q
c c
(D)

MS Page 8 of 47

JAM 2026 Mathematical Statistics (MS)

Q.10 Let " , " , … , "r , ) be independent and identically distributed A B, C

random variables. Define "s = ∑rX " , n = ∑rX " − "s .
r t
us 6v
p w q is
c
√ a
The distribution of

(A) Dt

(B) Dr

(C) Et

(D) Er

MS Page 9 of 47

JAM 2026 Mathematical Statistics (MS)

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

Q.11 Let : ℕ → 1, 2, 3, … , 100 be an onto function. Consider the following
statements.

P: If Z = sup 2 , 2 +2 , 2 + 4 ,… , ≥ 1, then
lim Z = lim sup .
→W →W

Q: If ℎ = inf , +1 , + 2 ,… , ≥ 1, then
lim ℎ 4 + 1 = lim inf .
→W →W

Then

(A) P is correct and Q is NOT correct

(B) P is NOT correct and Q is correct

(C) Both P and Q are correct

(D) Neither P nor Q is correct

MS Page 10 of 47

JAM 2026 Mathematical Statistics (MS)

Q.12 Let ℎ: ℝ → ℝ and Z: ℝ → ℝ be two bounded functions. Define the function
: ℝ → ℝ by
Z + ℎ , > 0,
=•
€ +^ , ≤ 0,
where €, ^ are real constants. Consider the following statements.

P: If € = Z 0 = ℎ 0 , then is differentiable at = 0.

, Z, and ℎ are thrice differentiable functions with Z 0 = 0 and
= 0.
Q: If
Z 0 + ℎ 0 > 0, then has a local minimum at

Then

(A) P is correct and Q is NOT correct

(B) P is NOT correct and Q is correct

(C) Both P and Q are correct

(D) Neither P nor Q is correct

MS Page 11 of 47

JAM 2026 Mathematical Statistics (MS)

Q.13 Consider the function : ℝ → ℝ given by
, d = d 5 + 3d − 6 .
Which one of the following statements is NOT true?

(A) has exactly 4 critical points

(B) has more than one saddle point

(C) has two local minima

has a critical point of the form $, + such that $ + + =
h
g
(D)

MS Page 12 of 47

JAM 2026 Mathematical Statistics (MS)

Q.14 Let : ℝ ∖ 0 → ℝ and Z: ℝ ∖ 0 → ℝ be two bounded and differentiable
functions. Define !: ℝ → ℝ by
d +d Z , d ≠ 0,
! ,d = •
0, d = 0.
Consider the following statements.

P: ! is continuous at 0, 0 .

ƒ„ ƒ„
exist at 1, 0 .
… …d
Q: Both the partial derivatives and

Then

(A) P is correct and Q is NOT correct

(B) P is NOT correct and Q is correct

(C) Both P and Q are correct

(D) Neither P nor Q is correct

MS Page 13 of 47

JAM 2026 Mathematical Statistics (MS)

Q.15 The bounded region enclosed by the curve d = + 2 and the line
d = 4 − 1 is revolved about the -axis to generate a solid. The volume of
the generated solid equals

(A) tt o
g

(B) tb o
g

(C) t o
g

(D) r o
g

MS Page 14 of 47

JAM 2026 Mathematical Statistics (MS)

Q.16 Let be an × real matrix. Consider the following statements.

P: If is skew-symmetric, then " " = 0, for every × 1 real matrix " .

Q: If is skew-symmetric and orthogonal, then is even.

Then

(A) P is correct and Q is NOT correct

(B) P is NOT correct and Q is correct

(C) Both P and Q are correct

(D) Neither P nor Q is correct

MS Page 15 of 47

JAM 2026 Mathematical Statistics (MS)

Q.17 Two different brands, namely Brand A and Brand B, of mobile batteries are
tested and the following data on their lifetimes (in months) are obtained:

Brand A 27 28 30 32 33

Brand B 17 19 20 21 23

Let &#† and &#‡ denote the sample coefficients of variation for Brand A and
Brand B, respectively. Let ˆ† and ˆ‡ denote the sample standard deviations for
Brand A and Brand B, respectively. Which one of the following statements is
true?

(A) &#† < &#‡ and ˆ† < ˆ‡

(B) &#† < &#‡ and ˆ† > ˆ‡

(C) &#† > &#‡ and ˆ† < ˆ‡

(D) &#† > &#‡ and ˆ† > ˆ‡

MS Page 16 of 47

JAM 2026 Mathematical Statistics (MS)

" " "c
Q.18 Consider the matrix

=‰ " " " c Š,
"c "c "cc
where " ‹ , 1 ≤ , Œ ≤ 3, are independent and identically distributed random

1
variables with the probability mass function

= 3 3, = −1, 0, 1,
0, otherwise.
The probability that is orthogonal equals

(A) e

c•

cŽ
(B)

(C) Ž

c•

(D) `

c•

MS Page 17 of 47

JAM 2026 Mathematical Statistics (MS)

Q.19 Let , ‘, & be three events such that & & > 0. Consider the
following statements.

S1: |& > ‘|& and |& > ‘|& together imply that
> ‘ .

S2: |& = |& implies that and & are independent.

Then

(A) S1 is correct and S2 is NOT correct

(B) S1 is NOT correct and S2 is correct

(C) Both S1 and S2 are correct

(D) Neither S1 nor S2 is correct

MS Page 18 of 47

JAM 2026 Mathematical Statistics (MS)

Q.20 Let " be a random variable having the moment generating function

’u E = 5 “ ”6
, E ∈ ℝ. Then, #$% 2" + 3 equals

(A) 16

(B) 4

(C) 6

(D) 8

MS Page 19 of 47

JAM 2026 Mathematical Statistics (MS)

If " is a random variable with the cumulative distribution function
0, ≤ 0,
Q.21

! =• 7–
1−5 6
– , > 0,
then " equals

—
(A) ˜

(B) √2o

—
(C)
˜

(D) 1

MS Page 20 of 47

JAM 2026 Mathematical Statistics (MS)

Q.22 Let the random variables " and ) denote the time to failure of component A
and component B, respectively, of an instrument. The joint probability density
function of ", ) is given by
1 6 f ™šc6_
, d = 3 10 5 , 5< < 10, d > − 3,
–

0, otherwise.
The probability, that component B fails before component A, is equal to

(A) “ √“ 6
“ √“

(B) “ √“
“ √“ š

(C) “ √“ 6
“ √“ š

“ √“
(D)

MS Page 21 of 47

JAM 2026 Mathematical Statistics (MS)

Q.23 Let " , " , … be a sequence of independent random variables with
1
" = = =1− " =0 , for = 1, 2, ….

Consider the following statements.

P: " converges in mean-square to 0.

Q: " converges in probability to 0.

Then

(A) P is correct and Q is NOT correct

(B) P is NOT correct and Q is correct

(C) Both P and Q are correct

(D) Neither P nor Q is correct

MS Page 22 of 47

JAM 2026 Mathematical Statistics (MS)

Q.24 Let " , " , … , " be a random sample of size ≥ 1 from a

1 c_ 6 œ7
distribution with the probability density function

= 5 •, ∈ ℝ,
2› c
where › > 0 is an unknown parameter. Which one of the following is a
maximum likelihood estimator of › ?

∑ X 5 už
(A) c

(B) ∑X "

∑ X 5 už
c
(C)

(D) ∑ X "c

MS Page 23 of 47

JAM 2026 Mathematical Statistics (MS)

Q.25 Let " , " , … , " a be a random sample of size 10 from an / ›
distribution, where › > 0 is an unknown parameter. Let Ÿ = ∑ Xa " .
a

Consider the following statements.

P: log “ 2 Ÿ is a maximum likelihood estimator of median of " .

Q: 5
–
" <2 .
6
¡ is a maximum likelihood estimator of

Then

(A) P is correct and Q is NOT correct

(B) P is NOT correct and Q is correct

(C) Both P and Q are correct

(D) Neither P nor Q is correct

MS Page 24 of 47

JAM 2026 Mathematical Statistics (MS)

Q.26 Let " , " , "c , "b be a random sample of size 4 from a D¢ distribution,
where £ ∈ ℕ is an unknown parameter. To test la : £ = 1 against
l : £ = 2, the critical region ∑bX " > 6 is being used. If I and ›
denote the probabilities of Type-I error and Type-II error, respectively, then
which one of the following statements is true?

[You may use D ,a.a bc = D ,a.abrt = Db,a. rr = Dt,a.hb¤ = 6]

(A) 0.20 < c I + › < 0.25
b b

(B) I > 0.20

(C) › < 0.30

(D) The power of the test lies in the interval p0, 1q
2

MS Page 25 of 47

JAM 2026 Mathematical Statistics (MS)

Q.27 Let " be a random sample of size 1 from a * 0, 0 distribution, where
0 > 0 is an unknown parameter. To test la : 0 ≥ 1 against l : 0 < 1, the
critical region " < is being used. If › ⋅ is the power function of the test,
c
b

then which one of the following statements is NOT true?

(A) The size of the test is c
b

inf › 0 =
c
¥¦ b
(B)

(C) For all 0 > 0, › 0 < 1

(D) lim › 0 = 1
¥→a

MS Page 26 of 47

JAM 2026 Mathematical Statistics (MS)

Q.28 The recorded air quality index (AQI) at a place in a city for 16 consecutive days
for the current year is compared with the historical average AQI. The direction
of the recorded AQI is noted as ‘+’ if it is above the historical average and as
‘−’ if it is below the historical average. The dataset is as below:
++−+−−++++−−−++−
The total number of runs, O , is being used to test
la : The direction of the recorded AQI is random
against
l : The direction of the recorded AQI is not random

Consider the following statements.

P: The observed value of total number of runs is 8.

Q: Based on the above dataset, la is rejected in favor of l at level 0.05.

Then which one of the following is true?

[You may use: Under la , with 7 negative signs and 9 positive signs,
O ≤ 6 = 0.108, O ≤ 7 = 0.231, O ≤ 8 = 0.427]

(A) P is correct and Q is NOT correct

(B) P is NOT correct and Q is correct

(C) Both P and Q are correct

(D) Neither P nor Q is correct

MS Page 27 of 47

JAM 2026 Mathematical Statistics (MS)

Q.29 Consider a discrete time Markov chain with state space n = 1, 2, 3 and the

1 1 1
transition probability matrix

⎡ ⎤
⎢2 4 4⎥
⎢0 1 4 ⎥.
⎢ 5 5⎥
⎣0 1 0⎦
Which one of the following statements is true?

(A) The Markov chain is irreducible

(B) 1 is a recurrent state

(C) 2 is a transient state

(D) 3 is a recurrent state

MS Page 28 of 47

JAM 2026 Mathematical Statistics (MS)

Q.30 Let " , " , "c be three independent random variables such that " ~ A p0, q,

" ~A 0, 2 , and "c ~A 0, 4 . Consider the following statements.

h uf–
has ! distribution with 1 and 2 degrees of freedom.
u–– šu`–
P:

uf 6u–
uf šu–
Q: has standard Cauchy distribution.

Then

(A) P is correct and Q is NOT correct

(B) P is NOT correct and Q is correct

(C) Both P and Q are correct

(D) Neither P nor Q is correct

MS Page 29 of 47

JAM 2026 Mathematical Statistics (MS)

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

Q.31 Let $ ®

$ +6
be a sequence of real numbers given by

$ = 1, $ š = , ≥ 1.
$ +2
Which of the following statements is/are true?

(A) $ a b − $ a h ≥ 0

(B) $ ®r is a monotone sequence

(C) $ ® is a convergent sequence

(D) $ ≤ 3 for all

MS Page 30 of 47

JAM 2026 Mathematical Statistics (MS)

Q.32 Let be an £ × real matrix with non-zero entries. Which of the following
statements is/are true?

(A) If £ < , then there exists a non-zero × 1 real matrix " such that
" = ¯, where ¯ is the £ × 1 zero matrix

(B) If £ > , then is a singular matrix

(C) If = 1 and £ > 1, then has nullity £ − 1

(D) If there exists an × £ matrix ‘ such that ‘ = ¢ and an ×£
matrix & such that & = , then =£

MS Page 31 of 47

JAM 2026 Mathematical Statistics (MS)

Q.33 The following data represent the lifetimes (in months) of a sample of eleven
bulbs:
15, 10, 12, 10, 17, 12, 14, 20, 13, 22, 15
Which of the following statements is/are true?

(A) The sample mean is greater than the sample median

(B) The mean deviation about the point 15 equals 3

(C) The range of the data is twice the interquartile range of the data

(D) The sample standard deviation as well as the sample median are scaled by 2 if

every data point is scaled by 2

MS Page 32 of 47

JAM 2026 Mathematical Statistics (MS)

Q.34 Let n = 1, 2, 3, … and suppose that every subset of n is an event. Let
° n denote the power set of n. Which of the following statements is/are
true?

(A) There exists a probability function : ° n → [0, ∞ such that
= , ≥1
š

(B) There exists a probability function : ° n → [0, ∞ such that
= 0 if is a finite set and = 1 if is an infinite set

(C) There exists a probability function : ° n → [0, ∞ such that
1, 2, … , =\ ^ , ≥1
_

(D) There exists a probability function : ° n → [0, ∞ such that
1, 2, … , = \a 5 6_ ^ , ≥1

MS Page 33 of 47

JAM 2026 Mathematical Statistics (MS)

Q.35 Let " be a random variable with the probability density function
1
⎧ , 0 < < 1,
⎪2
= 1
⎨ , 1 < < 3,
u
⎪4
⎩ 0, otherwise.

If ) = " and ¶ =
u
, then which of the following statements is/are true?

(A) A probability density function of ) is given by
1
⎧ , 0 < d < 1,
⎪ 4·d
v d = 1
⎨ , 1 < d < 9,
⎪ 8·d
⎩ 0, otherwise

(B) A probability density function of ¶ is given by
1 1
⎧ , < ¹ < 1,
⎪ 4¹ 3
¸ ¹ = 1
⎨ , 1 < ¹ < ∞,
⎪ 2¹
⎩ 0, otherwise

(C) ) < ∞ and ¶ <∞

(D) The distribution of ) is positively skewed

MS Page 34 of 47

JAM 2026 Mathematical Statistics (MS)

Q.36 Let ", ) be a random vector with the joint probability density function
5 6_ ™š , > 0, d > 0,
,d = •
0, otherwise.
Which of the following statements is/are true?

(A) The marginal distribution of " is / 1

(B) )|" = 4 = 0.25

(C) ") = 1.5

(D) ) is not finite

MS Page 35 of 47

JAM 2026 Mathematical Statistics (MS)

Q.37 Let " , " be a random sample of size 2 from an / 0 distribution,
where 0 > 0 is an unknown parameter. Let
1, "1 > 2, "1 + "2 − 2
Ÿ = • and Ÿ2 = max •0, ½ .
0, otherwise, "1 + "2
Which of the following statements is/are true?

(A) Ÿ is an unbiased estimator of 5 6 /¥

(B) Ÿ is the uniformly minimum variance unbiased estimator of 5 6 /¥

(C) #$% Ÿ ≤ 5 6b/¥ for all 0 > 0
¥

(D) " ," is a complete sufficient statistic

MS Page 36 of 47

JAM 2026 Mathematical Statistics (MS)

Q.38 Consider a simple linear regression model
) = ›a + › +¿ , = 1, 2, 3, 4, 5,
where the random errors ¿ have mean 0 and variance 36, and are

uncorrelated. Let ›Àa and ›À be the least squares estimators of ›a and › ,
respectively. The above model is fitted to the following dataset

−2 −1 0 1 2

d 2.7 $ −0.1 + −2.3

where $, + ∈ ℝ . Which of the following statements is/are true?

(A) If the observed value of ›Àa , ›À is 0, −0.11 , then $, + = −4.6, 4.3

(B) If $, + = 0, 1 , then the observed value of ›Àa , ›À is 0.26, 0.20

s=
s , ›Àa H = 7.2, where )
(C) &'( G) ∑gX )
g

(D) #$%G›Àa H = 2 #$% ›À

MS Page 37 of 47

JAM 2026 Mathematical Statistics (MS)

Q.39 Let ) , ) be a random sample of size 2 from a distribution with the

0 1 − 0 ™ , d = 0, 1, 2, …
probability mass function

d = •
0, otherwise,
where 0 ∈ 0, 1 is an unknown parameter. Let Á be the most powerful
randomized test for testing la ∶ 0 = 0.20 against l ∶ 0 = 0.75 at level
0.05. Which of the following statements is/are true?

(A) The test Á rejects la with probability 1 if the observed value of

) + ) is 0

(B) The test Á rejects la with probability 1 if the observed value of

) + ) is 1

(C) The test Á rejects l with probability g if the observed value of
a h

) ) is 1

(D) The power of the test Á is h
a b

MS Page 38 of 47

JAM 2026 Mathematical Statistics (MS)

Q.40 Let " , " , … , " be a random sample of size > 1 from an /›
distribution, where › > 0. If Ÿ = ∑ X " , then which of the following
statements is/are true?

(A) Ÿ follows a gamma distribution with mean − 1 › and variance
−1 ›

(B) 2Ÿ
follows a D
Ã
distribution

(C) ›Ÿ Ö Ãe
b
follows a gamma distribution with mean and variance

(D) The mode of Ÿ is −1 ›

MS Page 39 of 47

JAM 2026 Mathematical Statistics (MS)

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

Q.41 Let : ℝ → ℝ be the function defined by = 3 5 _ cos 2 .
If / =$ c
++ +€ + ^ is the third degree Taylor polynomial
of at = 0, then |$| + |+| + |€ | + |^ | equals __________ (in integer)

+d+¹ = 1
Q.42 If the system of linear equations

+2d−¹ = +
$ +7d+¹ = + + 1,
where $ and + are real constants, has infinitely many solutions, then $ + +
equals __________ (in integer)

Q.43 In a singing competition, two judges award ranks for each of the 10 contestants.

coefficient Å for the 10 contestants was computed to be 0.6. It was found later
Based on the ranks awarded by the two judges, the Spearman’s rank correlation

that the difference in ranks awarded by the two judges for one of the contestant
was incorrectly taken as 2, instead of the correct value 5. If the correct difference
is taken into account, then the corrected value of Å equals __________ (rounded
off to two decimal places)

MS Page 40 of 47

JAM 2026 Mathematical Statistics (MS)

Q.44 A point is chosen at random from a square region whose sides are of 2 metres.
If the centre of the square is defined to be the point of intersection of its two
diagonals, then the probability that the randomly chosen point is closer to the
centre of the square than to any of its four corners equals __________ (rounded
off to two decimal places)

Q.45 Let " be a continuous random variable with mean 2 and variance 4. Using
Chebyshev’s inequality, the lower bound for −4 ≤ " ≤ 8 equals
__________ (rounded off to two decimal places)

Q.46 Let ", ) be a random vector having the bivariate normal distribution with
" = 2, ) = 10, #$% " = 9, #$% ) = 25, and the correlation
coefficient between " and ) equals Å > 0.
If 4 < ) < 16 |" = 2 = 0.8664, then Å equals __________ (rounded
off to two decimal places)
[You may use Φ 1.5 = 0.9332, Φ 2 = 0.9772]

MS Page 41 of 47

JAM 2026 Mathematical Statistics (MS)

Q.47 Let " ® be a sequence of independent and identically distributed random

variables having * 0, 3 distribution. If ) = ∑ X " , ≥ 1, then

) ® converges in probability to __________ (in integer)

Q.48 Let " , " , "c , "b be a random sample of size 4 from a distribution with the
probability density function
I I+1 F6
1− , 0 < < 1,
= •
0, otherwise,
where I > 0 is an unknown parameter. Based on the observed data
0.2, 0.1, 0.3, 0.4, the method of moments estimate of I equals __________
(rounded off to two decimal places)

Q.49 Let A E , E ≥ 0 be a Poisson process with rate Æ = 1. Then
[A 8 A 5 ] equals __________ (in integer)

MS Page 42 of 47

JAM 2026 Mathematical Statistics (MS)

Q.50 Let " , " , "c be a random sample of size 3 from * 1, 2 distribution.
Then " c > 1.5 equals __________ (rounded off to two decimal places)

MS Page 43 of 47

JAM 2026 Mathematical Statistics (MS)

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

Q.51 The area of the region in the first quadrant that is bounded by the parabola
d = 36 , the line d = 3 − 9 and -axis equals __________ (in integer)

Q.52 Consider the real matrices

$ 3^ 3 $ + € $ + €
=‰+ 35 −2 Š , ‘ = ‰ 1 −3 5 Š, & = ‰ ^ 5 Š.
€ 3 1 2^ 25 2 −1 −4 9

If the determinant of is − 48 and the determinant of ‘ is 2, then the
determinant of & equals __________ (in integer)

Q.53 A fair six-sided die is rolled 4 times independently. If / is the probability that
the sum of the outcomes is 14, then 10/ equals __________ (rounded off to
three decimal places)

MS Page 44 of 47

JAM 2026 Mathematical Statistics (MS)

Q.54 Let " be a random variable with the probability density function
3
=34 2− , 0< < 2,
0, otherwise.
If ¶ = |" − 1| and £ is the median of ¶, then 12 £ − 4 £c equals
__________ (in integer)

Q.55 If ", ) is a random vector with the joint probability mass function

3 _6™6
, d = È25 , = 0, 1, 2, … ; d = , + 1, + 2, … ,
6c
!
0, otherwise,
then #$% )|" = 2 equals __________ (rounded off to two decimal places)

Q.56 Let " , " , … be a sequence of independent and identically distributed random

variables with mean 4 and variance 9. If Ÿ = ∑ X " , then

lim 4√ − 3 < √ Ÿ < 6 + 4√
→W
equals __________ (rounded off to

two decimal places)

[You may use Φ 2 = 0.9772, Φ 1.5 = 0.9332, Φ 1 = 0.8413]

MS Page 45 of 47

JAM 2026 Mathematical Statistics (MS)

Q.57 Let " , " , … , " be a random sample of size > 1 from a A B, 1
distribution, where B ∈ ℝ is an unknown parameter. If "s = ∑ X " , then

the minimum value of such that 2"s − 1 < 2B < 2 "s + 1 ≥ 0.99
equals __________ (in integer)

[You may use Φ 2.58 = 0.9951, Φ 2.48 = 0.9934]

Q.58 Let " , " be a random sample of size 2 from a distribution with the
probability density function
I F6 , 0 < < 1,
=•
0, otherwise,
where I > 0 is an unknown parameter. To test la : I = 1 against
l : I = 2, the most powerful test is being used. If the observed value of the

, , then the /-value of the test equals __________
b
random sample is

(rounded off to three decimal places)

MS Page 46 of 47

JAM 2026 Mathematical Statistics (MS)

Q.59 The observed value of a random sample of size 9 from a distribution having
continuous and strictly increasing cumulative distribution function is as below:
1.00, −0.01, 0.70, 0.25, −1.00, 1.20, −0.33, 0.68, −2.00
Let ’ be the unknown median of the distribution. To test la : ’ = 0 against
l : ’ > 0, the sign test is being used. If
1, if la is accepted at level 0.05,
Ë=•
0, if la is rejected at level 0.05,
and / is the /-value of the test, then based on the above data / + Ë equals
__________ (rounded off to two decimal places)

Q.60 Let " , " , "c be a random sample of size 3 from / 1 distribution.
Then, G4" 5 u – H equals __________ (in integer)

END OF THE QUESTION PAPER

MS Page 47 of 47