IIT JAM Mathematical Statistics - MS 2022 | Sourav Sir's Classes

JAM 2022 MATHEMATICAL STATISTICS - MS
MS 2/52
Special Instructions / Useful Data
Special Instructions / Useful Data
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
𝐼𝑛 𝑛 × 𝑛 identity matrix, 𝑛 = 2, 3, 4, …
𝐸𝑐 Complement of a set 𝐸
𝑃(𝐸) Probability of an event 𝐸
𝑃(𝐸 | 𝐹) Conditional probability of an event 𝐸 given the occurrence of the event 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 < ∞
𝐸𝑥𝑝(𝜆) Exponential distribution with the probability density function, for 𝜆 > 0,
𝑓(𝑥) = {
𝜆 𝑒− 𝜆𝑥
, 𝑥 > 0
0, otherwise
𝑁(𝜇, 𝜎2
) Normal distribution with mean 𝜇 and variance 𝜎2
, 𝜇 ∈ ℝ, 𝜎 > 0
Φ(⋅) The cumulative distribution function of 𝑁(0, 1) distributed random variable
𝜒𝑛
2 Central chi-square distribution with 𝑛 degrees of freedom, 𝑛 = 1, 2, …
𝐹
𝑚,𝑛 Snedecor’s central 𝐹-distribution with (𝑚, 𝑛) degrees of freedom,
𝑚, 𝑛 = 1, 2, …
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)
Φ(1.645) = 0.95, Φ(0.355) = 0.6387
𝑡8,0.0185 = 2.5

MS 3/52
Section A: Q.1 – Q.10 Carry ONE mark each.
Q.1 Let {𝑎𝑛}𝑛≥1 be a sequence of non-zero real numbers. Then which one of the
following statements is true?
(A)
If {
𝑎𝑛+1
𝑎𝑛
}
𝑛≥1
is a convergent sequence, then {𝑎𝑛}𝑛≥1 is also a convergent
sequence
(B) If {𝑎𝑛}𝑛≥1 is a bounded sequence, then {𝑎𝑛}𝑛≥1 is a convergent sequence
(C) If |𝑎𝑛+2 − 𝑎𝑛+1| ≤
3
4
|𝑎𝑛+1 − 𝑎𝑛| for all 𝑛 ≥ 1, then {𝑎𝑛}𝑛≥1 is a Cauchy
sequence
(D) If {|𝑎𝑛|}𝑛≥1 is a Cauchy sequence, then {𝑎𝑛}𝑛≥1 is also a Cauchy sequence

MS 4/52
Q.2 Let 𝑓: ℝ → ℝ be the function defined by
𝑓(𝑥) = {lim
ℎ→0
(𝑥 + ℎ) sin (
1
𝑥
+ ℎ) − 𝑥 sin
1
𝑥
ℎ
, 𝑥 ≠ 0
0, 𝑥 = 0.
Then which one of the following statements is NOT true?
(A) 𝑓 (
2
𝜋
) = 1
(B) 𝑓 (
1
𝜋
) =
1
𝜋
(C) 𝑓 (−
2
𝜋
) = −1
(D) 𝑓 is not continuous at 𝑥 = 0

MS 5/52
Q.3 Let 𝑓: ℝ → ℝ be the function defined by
𝑓(𝑥) = det (
1 + 𝑥 9 9
9 1 + 𝑥 9
9 9 1 + 𝑥
) .
Then the maximum value of 𝑓 on the interval [9, 10] equals
(A) 118
(B) 112
(C) 114
(D) 116
Q.4 Let 𝐴 and 𝐵 be two events such that 0 < 𝑃(𝐴) < 1 and 0 < 𝑃(𝐵) < 1.
(A) If 𝑃(𝐴|𝐵) > 𝑃(𝐴), then 𝑃(𝐵|𝐴) > 𝑃(𝐵)
(B) If 𝑃(𝐴 ∪ 𝐵) = 1, then 𝐴 and 𝐵 cannot be independent
(C) If 𝑃(𝐴|𝐵) > 𝑃(𝐴), then 𝑃(𝐴𝑐|𝐵) < 𝑃(𝐴𝑐)
(D) If 𝑃(𝐴|𝐵) > 𝑃(𝐴), then 𝑃(𝐴𝑐|𝐵𝑐) < 𝑃(𝐴𝑐)

MS 6/52
Q.5 If 𝑀(𝑡), 𝑡 ∈ ℝ, is the moment generating function of a random variable, then
which one of the following is NOT the moment generating function of any
random variable?
(A) 5𝑒−5𝑡
1−4𝑡2
𝑀(𝑡), |𝑡| <
1
2
(B) 𝑒−𝑡
𝑀(𝑡), 𝑡 ∈ ℝ
(C) 1+𝑒𝑡
2(2−𝑒𝑡)
𝑀(𝑡), 𝑡 < ln 2
(D) 𝑀(4𝑡), 𝑡 ∈ ℝ
Q.6 Let 𝑋 be a random variable having binomial distribution with parameters
𝑛 (> 1) and 𝑝 (0 < 𝑝 < 1). Then 𝐸 (
1
1+𝑋
) equals
(A) 1−(1−𝑝)𝑛+1
(𝑛+1)𝑝
(B) 1−𝑝𝑛+1
(𝑛+1)(1−𝑝)
(C) (1−𝑝)𝑛+1
𝑛(1−𝑝)
(D) 1−𝑝𝑛
(𝑛+1)𝑝

MS 7/52
Q.7 Let (𝑋, 𝑌) be a random vector having the joint probability density function
𝑓(𝑥, 𝑦) = {
√2
√𝜋
𝑒−2𝑥
𝑒−
(𝑦−𝑥)2
2 , 0 < 𝑥 < ∞, −∞ < 𝑦 < ∞
0, otherwise.
Then 𝐸(𝑌) equals
(A) 1
2
(B) 2
(C) 1
(D) 1
4

MS 8/52
Q.8 Let 𝑋1 and 𝑋2 be two independent and identically distributed discrete
random variables having the probability mass function
𝑓(𝑥) = {(
1
2
)
𝑥
, 𝑥 = 1, 2, 3, …
0, otherwise.
Then 𝑃(min{𝑋1, 𝑋2} ≥ 5) equals
(A) 1
256
(B) 1
512
(C) 1
64
(D) 9
256

MS 9/52
Q.9 Let 𝑋1, 𝑋2, … , 𝑋𝑛 (𝑛 ≥ 2) be a random sample from 𝐸𝑥𝑝 (
1
𝜃
) distribution,
where 𝜃 > 0 is unknown. If 𝑋 =
1
𝑛
∑ 𝑋𝑖
𝑛
𝑖=1 , then which one of the following
statements is NOT true?
(A) 𝑋 is the uniformly minimum variance unbiased estimator of 𝜃
(B)
𝑋
2
is the uniformly minimum variance unbiased estimator of 𝜃2
(C) 𝑛
𝑛+1
𝑋
2
is the uniformly minimum variance unbiased estimator of 𝜃2
(D) 𝑉𝑎𝑟 (𝐸(𝑋𝑛 | 𝑋 )) ≤ 𝑉𝑎𝑟(𝑋𝑛)

MS 10/52
Q.10 Let 𝑋1, 𝑋2, … , 𝑋𝑛 (𝑛 ≥ 3) be a random sample from a 𝑁(𝜇, 𝜎2
)
distribution, where 𝜇 ∈ ℝ and 𝜎 > 0 are both unknown. Then which one of
the following is a simple null hypothesis?
(A) 𝐻0: 𝜇 < 5, 𝜎2
= 3
(B) 𝐻0: 𝜇 = 5, 𝜎2
> 3
(C) 𝐻0: 𝜇 = 5, 𝜎2
= 3
(D) 𝐻0: 𝜇 = 5
Section A: Q.11 – Q.30 Carry TWO marks each.
Q.11
lim
𝑛→∞
6
𝑛+2
{(2 +
1
𝑛
)
2
+ (2 +
2
𝑛
)
2
+ ⋯ + (2 +
𝑛−1
𝑛
)
2
} equals
(A) 38
(B) 36
(C) 32
(D) 30

MS 11/52
Q.12 Let 𝑓: ℝ2
→ ℝ be the function defined by
𝑓(𝑥, 𝑦) = { 𝑥2
sin
1
𝑥
+ 𝑦2
cos 𝑦 , 𝑥 ≠ 0
0, 𝑥 = 0.
(A) 𝑓 is continuous at (0, 0)
(B) The partial derivative of 𝑓 with respect to 𝑥 is not continuous at (0, 0)
(C) The partial derivative of 𝑓 with respect to 𝑦 is continuous at (0, 0)
(D) 𝑓 is not differentiable at (0, 0)

MS 12/52
Q.13 Let 𝑓: [1, 2] → ℝ be the function defined by
𝑓(𝑡) = ∫ √𝑥2𝑒𝑥2
− 1
𝑡
1
𝑑𝑥.
Then the arc length of the graph of 𝑓 over the interval [1, 2] equals
(A) 𝑒2
− √𝑒
(B) 𝑒 − √𝑒
(C) 𝑒2
− 𝑒
(D) 𝑒2
− 1

MS 13/52
Q.14 Let 𝐹: [0, 2] → ℝ be the function defined by
𝐹(𝑥) = ∫ 𝑒𝑥 [𝑡]
𝑑𝑡,
𝑥+2
𝑥2
where [𝑡] denotes the greatest integer less than or equal to 𝑡. Then the value
of the derivative of 𝐹 at 𝑥 = 1 equals
(A) 𝑒3
+ 2𝑒2
− 𝑒
(B) 𝑒3
− 𝑒2
+ 2𝑒
(C) 𝑒3
− 2𝑒2
+ 𝑒
(D) 𝑒3
+ 2𝑒2
+ 𝑒

MS 14/52
Q.15 Let the system of equations
𝑥 + 𝑎𝑦 + 𝑧 = 1
2𝑥 + 4𝑦 + 𝑧 = − 𝑏
3𝑥 + 𝑦 + 2𝑧 = 𝑏 + 2
have infinitely many solutions, where 𝑎 and 𝑏 are real constants. Then the
value of 2𝑎 + 8𝑏 equals
(A) −11
(B) −10
(C) −13
(D) −14

MS 15/52
Q.16
Let 𝐴 = (
0 1 0
1 0 0
0 1 1
). Then the sum of all the elements of 𝐴100
equals
(A) 101
(B) 103
(C) 102
(D) 100

MS 16/52
Q.17 Suppose that four persons enter a lift on the ground floor of a building. There
are seven floors above the ground floor and each person independently chooses
her exit floor as one of these seven floors. If each of them chooses the topmost
floor with probability
1
3
and each of the remaining floors with an equal
probability, then the probability that no two of them exit at the same floor
equals
(A) 200
729
(B) 220
729
(C) 240
729
(D) 180
729

MS 17/52
Q.18 A year is chosen at random from the set of years {2012, 2013, … , 2021}.
From the chosen year, a month is chosen at random and from the chosen month,
a day is chosen at random. Given that the chosen day is the 29th
of a month,
the conditional probability that the chosen month is February equals
(A) 279
9965
(B) 289
9965
(C) 269
9965
(D) 259
9965

MS 18/52
Q.19 Suppose that a fair coin is tossed repeatedly and independently. Let 𝑋 denote
the number of tosses required to obtain for the first time a tail that is
immediately preceded by a head. Then 𝐸(𝑋) and 𝑃(𝑋 > 4), respectively,
are
(A) 4 and
5
16
(B) 4 and
11
16
(C) 6 and
5
16
(D) 6 and
11
16

MS 19/52
Q.20 Let 𝑋 be a random variable with the moment generating function
𝑀(𝑡) =
1
(1 − 4𝑡)5
, 𝑡 <
1
4
.
Then the lower bounds for 𝑃(𝑋 < 40), using Chebyshev’s inequality and
Markov’s inequality, respectively, are
(A) 4
5
and
1
2
(B) 5
6
and
1
2
(C) 4
5
and
5
6
(D) 5
6
and
5
6

MS 20/52
Q.21 In a store, the daily demand for milk (in litres) is a random variable having
𝐸𝑥𝑝(𝜆) distribution, where 𝜆 > 0. At the beginning of the day, the store
purchases 𝑐 (> 0) litres of milk at a fixed price 𝑏 (> 0) per litre. The milk
is then sold to the customers at a fixed price 𝑠 (> 𝑏) per litre. At the end of
the day, the unsold milk is discarded. Then the value of 𝑐 that maximizes the
expected net profit for the store equals
(A)
−
1
𝜆
ln (
𝑏
𝑠
)
(B)
−
1
𝜆
ln (
𝑏
𝑠+𝑏
)
(C)
−
1
𝜆
ln (
𝑠−𝑏
𝑠
)
(D)
−
1
𝜆
ln (
𝑠
𝑠+𝑏
)

MS 21/52
Q.22 Let 𝑋1, 𝑋2 and 𝑋3 be three independent and identically distributed random
variables having 𝑈(0, 1) distribution. Then 𝐸 [(
ln 𝑋1
ln 𝑋1𝑋2𝑋3
)
2
] equals
(A) 1
6
(B) 1
3
(C) 1
8
(D) 1
4

MS 22/52
Q.23 Let (𝑋, 𝑌) be a random vector having bivariate normal distribution with
parameters 𝐸(𝑋) = 0, 𝑉𝑎𝑟(𝑋) = 1, 𝐸(𝑌) = −1, 𝑉𝑎𝑟(𝑌) = 4 and
𝜌(𝑋, 𝑌) = −
1
2
, where 𝜌(𝑋, 𝑌) denotes the correlation coefficient between
𝑋 and 𝑌. Then 𝑃(𝑋 + 𝑌 > 1 | 2𝑋 − 𝑌 = 1) equals
(A) Φ (−
1
2
)
(B) Φ (−
1
3
)
(C) Φ (−
1
4
)
(D) Φ (−
4
3
)

MS 23/52
Q.24 Let {𝑋𝑛}𝑛≥1 be a sequence of independent and identically distributed random
variables having the common probability density function
𝑓(𝑥) = {
2
𝑥3
, 𝑥 ≥ 1
0, otherwise.
If lim
𝑛→∞
𝑃 (|
1
𝑛
∑ 𝑋𝑖
𝑛
𝑖=1 − 𝜃| < 𝜖) = 1 for all 𝜖 > 0, then 𝜃 equals
(A) 4
(B) 2
(C) ln 4
(D) ln 2

MS 24/52
Q.25 Let 0.2, 1.2, 1.4, 0.3, 0.9, 0.7 be the observed values of a random sample of
size 6 from a continuous distribution with the probability density function
𝑓(𝑥) =
{
1, 0 < 𝑥 ≤
1
2
1
2𝜃 − 1
,
1
2
< 𝑥 ≤ 𝜃
0, otherwise,
where 𝜃 >
1
2
is unknown. Then the maximum likelihood estimate and the
method of moments estimate of 𝜃, respectively, are
(A) 7
5
and 2
(B) 47
60
and
32
15
(C) 7
5
and
32
15
(D) 7
5
and
47
60

MS 25/52
Q.26 For 𝑛 = 1, 2, 3, …, let the joint moment generating function of (𝑋, 𝑌𝑛) be
𝑀𝑋,𝑌𝑛
(𝑡1, 𝑡2) = 𝑒
𝑡1
2
2 (1 − 2𝑡2)−
𝑛
2 , 𝑡1 ∈ ℝ, 𝑡2 <
1
2
.
If 𝑇𝑛 =
√𝑛 𝑋
√𝑌𝑛
, 𝑛 ≥ 1, then which one of the following statements is true?
(A) The minimum value of 𝑛 for which 𝑉𝑎𝑟(𝑇𝑛) is finite is 2
(B) 𝐸(𝑇10
3 ) = 10
(C) 𝑉𝑎𝑟(𝑋 + 𝑌4) = 7
(D)
lim
𝑛→∞
𝑃(|𝑇𝑛| > 3) = 1 −
√2
√𝜋
∫ 𝑒−
𝑡2
2 𝑑𝑡
3
0

MS 26/52
Q.27 Let 𝑋(1) < 𝑋(2) < ⋯ < 𝑋(9) be the order statistics corresponding to a
random sample of size 9 from 𝑈(0, 1) distribution. Then which one of the
following statements is NOT true?
(A)
𝐸 (
𝑋(9)
1−𝑋(9)
) is finite
(B) 𝐸(𝑋(5)) = 0.5
(C) The median of 𝑋(5) is 0.5
(D) The mode of 𝑋(5) is 0.5

MS 27/52
Q.28 Let 𝑋1, 𝑋2, … , 𝑋16 be a random sample from a 𝑁(4𝜇, 1) distribution and
𝑌1, 𝑌2, … , 𝑌8 be a random sample from a 𝑁(𝜇, 1) distribution, where
𝜇 ∈ ℝ is unknown. Assume that the two random samples are independent.
If you are looking for a confidence interval for 𝜇 based on the statistic
8𝑋 + 𝑌, where 𝑋 =
1
16
∑ 𝑋𝑖
16
𝑖=1 and 𝑌 =
1
8
∑ 𝑌𝑖
8
𝑖=1 , then which one of the
following statements is true?
(A) There exists a 90% confidence interval for 𝜇 of length less than 0.1
(B) There exists a 90% confidence interval for 𝜇 of length greater than 0.3
(C)
[
8𝑋+𝑌
33
−
1.645
2√66
,
8𝑋+𝑌
33
+
1.645
2√66
] is the unique 90% confidence interval for 𝜇
(D) 𝜇 always belongs to its 90% confidence interval

MS 28/52
Q.29 Let 𝑋1, 𝑋2, 𝑋3, 𝑋4 be a random sample from a distribution with the probability
mass function
𝑓(𝑥) = {
𝜃𝑥(1 − 𝜃)1−𝑥
, 𝑥 = 0, 1
0, otherwise,
where 𝜃 ∈ (0, 1) is unknown. Let 0 < 𝛼 ≤ 1. To test the hypothesis
𝐻0: 𝜃 =
1
2
against 𝐻1: 𝜃 >
1
2
, consider the size 𝛼 test that rejects 𝐻0 if and
only if ∑ 𝑋𝑖
4
𝑖=1 ≥ 𝑘𝛼, for some 𝑘𝛼 ∈ {0, 1, 2, 3, 4}. Then for which one of
the following values of 𝛼, the size 𝛼 test does NOT exist?
(A) 1
16
(B) 1
4
(C) 11
16
(D) 5
16

MS 29/52
Q.30 Let 𝑋1, 𝑋2, 𝑋3, 𝑋4 be a random sample from a Poisson distribution with
unknown mean 𝜆 > 0. For testing the hypothesis
𝐻0: 𝜆 = 1 against 𝐻1: 𝜆 = 1.5,
let 𝛽 denote the power of the test that rejects 𝐻0 if and only if ∑ 𝑋𝑖
4
𝑖=1 ≥ 5.
Then which one of the following statements is true?
(A) 𝛽 > 0.80
(B) 0.75 < 𝛽 ≤ 0.80
(C) 0.70 < 𝛽 ≤ 0.75
(D) 0.65 < 𝛽 ≤ 0.70

MS 30/52
Section B: Q.31 – Q.40 Carry TWO marks each.
Q.31 Let {𝑎𝑛}𝑛≥1 be a sequence of real numbers such that 𝑎𝑛 =
1
3𝑛
for all 𝑛 ≥ 1.
Then which of the following statements is/are true?
(A) ∑ (−1)𝑛+1
𝑎𝑛
∞
𝑛=1 is a convergent series
(B) ∑
(−1)𝑛+1
𝑛
∞
𝑛=1 (𝑎1 + 𝑎2 + … + 𝑎𝑛) is a convergent series
(C)
The radius of convergence of the power series ∑ 𝑎𝑛𝑥𝑛
∞
𝑛=1 is
1
3
(D) ∑ 𝑎𝑛 sin
1
𝑎𝑛
∞
𝑛=1 is a convergent series

MS 31/52
Q.32 Let 𝑓: ℝ2
→ ℝ be the function defined by
𝑓(𝑥, 𝑦) = 8(𝑥2
− 𝑦2) − 𝑥4
+ 𝑦4
.
Then which of the following statements is/are true?
(A) 𝑓 has 9 critical points
(B) 𝑓 has a saddle point at (2, 2)
(C) 𝑓 has a local maximum at (−2, 0)
(D) 𝑓 has a local minimum at (0, −2)
Q.33 If 𝑛 ≥ 2, then which of the following statements is/are true?
(A) If 𝐴 and 𝐵 are 𝑛 × 𝑛 real orthogonal matrices such that
det(𝐴) + det(𝐵) = 0, then 𝐴 + 𝐵 is a singular matrix
(B) If 𝐴 is an 𝑛 × 𝑛 real matrix such that 𝐼𝑛 + 𝐴 is non-singular, then
𝐼𝑛 + (𝐼𝑛 + 𝐴)−1
(𝐼𝑛 − 𝐴) is a singular matrix
(C) If 𝐴 is an 𝑛 × 𝑛 real skew-symmetric matrix, then 𝐼𝑛 − 𝐴2
is a non-
singular matrix
(D) If 𝐴 is an 𝑛 × 𝑛 real orthogonal matrix, then det(𝐴 − 𝜆𝐼𝑛) ≠ 0 for all
𝜆 ∈ {𝑥 ∈ ℝ ∶ 𝑥 ≠ ±1}

MS 32/52
Q.34 Let Ω = {1, 2, 3, … } be the sample space of a random experiment and
suppose that all subsets of Ω are events. Further, let 𝑃 be a probability
function such that 𝑃({𝑖}) > 0 for all 𝑖 ∈ Ω. Then which of the following
statements is/are true?
(A) For every 𝜖 > 0, there exists an event 𝐴 such that 0 < 𝑃(𝐴) < 𝜖
(B) There exists a sequence of disjoint events {𝐴𝑘}𝑘≥1 with 𝑃(𝐴𝑘) ≥ 10−6
for
all 𝑘 ≥ 1
(C) There exists 𝑗 ∈ Ω such that 𝑃({𝑗}) ≥ 𝑃({𝑖}) for all 𝑖 ∈ Ω
(D) Let {𝐴𝑘}𝑘≥1 be a sequence of events such that ∑ 𝑃(𝐴𝑘)
∞
𝑘=1 < ∞.
Then for each 𝑖 ∈ Ω there exists 𝑁 ≥ 1 (which may depend on 𝑖)
such that 𝑖 ∉ ⋃ 𝐴𝑘
∞
𝑘=𝑁

MS 33/52
Q.35 A university bears the yearly medical expenses of each of its employees up to a
maximum of Rs. 1000. If the yearly medical expenses of an employee exceed
Rs. 1000, then the employee gets the excess amount from an insurance policy
up to a maximum of Rs. 500. If the yearly medical expenses of a randomly
selected employee has 𝑈(250, 1750) distribution and 𝑌 denotes the amount
the employee gets from the insurance policy, then which of the following
(A) 𝐸(𝑌) =
500
3
(B) 𝑃(𝑌 > 300) =
3
10
(C) The median of 𝑌 is zero
(D) The quantile of order 0.6 for 𝑌 equals 100

MS 34/52
Q.36 Let 𝑋 and 𝑌 be two independent random variables having 𝑁(0, 𝜎1
2
) and
𝑁(0, 𝜎2
2
) distributions, respectively, where 0 < 𝜎1 < 𝜎2. Then which of the
following statements is/are true?
(A) 𝑋 + 𝑌 and 𝑋 − 𝑌 are independent
(B) 2𝑋 + 𝑌 and 𝑋 − 𝑌 are independent if 2𝜎1
2
= 𝜎2
2
(C) 𝑋 + 𝑌 and 𝑋 − 𝑌 are identically distributed
(D) 𝑋 + 𝑌 and 2𝑋 − 𝑌 are independent if 2𝜎1
2
= 𝜎2
2

MS 35/52
Q.37 Let (𝑋, 𝑌) be a discrete random vector. Then which of the following
(A) If 𝑋 and 𝑌 are independent, then 𝑋2
and |𝑌| are also independent.
(B) If the correlation coefficient between 𝑋 and 𝑌 is 1, then
𝑃(𝑌 = 𝑎𝑋 + 𝑏) = 1 for some 𝑎, 𝑏 ∈ ℝ
(C) If 𝑋 and 𝑌 are independent and 𝐸[(𝑋𝑌)2] = 0, then 𝑃(𝑋 = 0) = 1 or
𝑃(𝑌 = 0) = 1
(D) If 𝑉𝑎𝑟(𝑋) = 0, then 𝑋 and 𝑌 are independent

MS 36/52
Q.38 Let 𝑋1, 𝑋2 and 𝑋3 be three independent and identically distributed random
variables having 𝑁(0, 1) distribution. If
𝑈 =
2𝑋1
2
(𝑋2 + 𝑋3)2
and 𝑉 =
2(𝑋2 − 𝑋3)2
2𝑋1
2
+ (𝑋2 + 𝑋3)2
,
then which of the following statements is/are true?
(A) 𝑈 has 𝐹1,1 distribution and 𝑉 has 𝐹1,2 distribution
(B) 𝑈 has 𝐹1,1 distribution and 𝑉 has 𝐹2,1 distribution
(C) 𝑈 and 𝑉 are independent
(D) 1
2
𝑉(1 + 𝑈) has 𝐹2,3 distribution

MS 37/52
Q.39 Let 𝑋1, 𝑋2, 𝑋3, 𝑋4 be a random sample from a continuous distribution with
the probability density function 𝑓(𝑥) =
1
2
𝑒−|𝑥−𝜃|
, 𝑥 ∈ ℝ, where 𝜃 ∈ ℝ is
unknown. Let the corresponding order statistics be denoted by
𝑋(1) < 𝑋(2) < 𝑋(3) < 𝑋(4). Then which of the following statements is/are
true?
(A) 1
2
(𝑋(2) + 𝑋(3)) is the unique maximum likelihood estimator of 𝜃
(B) (𝑋(1), 𝑋(2), 𝑋(3), 𝑋(4)) is a sufficient statistic for 𝜃
(C) 1
4
(𝑋(2) + 𝑋(3)) (𝑋(2) + 𝑋(3) + 2) is a maximum likelihood estimator of
𝜃(𝜃 + 1)
(D) (𝑋1𝑋2𝑋3, 𝑋1𝑋2𝑋4) is a complete statistic

MS 38/52
Q.40 Let 𝑋1, 𝑋2, … , 𝑋𝑛 (𝑛 > 1) be a random sample from a 𝑁(𝜇, 1) distribution,
where 𝜇 ∈ ℝ is unknown. Let 0 < 𝛼 < 1. To test the hypothesis 𝐻0: 𝜇 = 0
against 𝐻1: 𝜇 = 𝛿, where 𝛿 > 0 is a constant, let 𝛽 denote the power of the
size 𝛼 test that rejects 𝐻0 if and only if
1
𝑛
∑ 𝑋𝑖 > 𝑐𝛼
𝑛
𝑖=1 , for some constant
𝑐𝛼. Then which of the following statements is/are true?
(A) For a fixed value of 𝛿, 𝛽 increases as 𝛼 increases
(B) For a fixed value of 𝛼, 𝛽 increases as 𝛿 increases
(C) For a fixed value of 𝛿, 𝛽 decreases as 𝛼 increases
(D) For a fixed value of 𝛼, 𝛽 decreases as 𝛿 increases

MS 39/52
Section C: Q.41 – Q.50 Carry ONE mark each.
Q.41 Let {𝑎𝑛}𝑛≥1 be a sequence of real numbers such that 𝑎1+5𝑚 = 2,
𝑎2+5𝑚 = 3, 𝑎3+5𝑚 = 4, 𝑎4+5𝑚 = 5, 𝑎5+5𝑚 = 6, 𝑚 = 0, 1, 2, … . Then
limsupn→∞ 𝑎𝑛 + liminfn→∞ 𝑎𝑛 equals __________
Q.42 Let 𝑓: ℝ → ℝ be a function such that
20(𝑥 − 𝑦) ≤ 𝑓(𝑥) − 𝑓(𝑦) ≤ 20(𝑥 − 𝑦) + 2(𝑥 − 𝑦)2
for all 𝑥, 𝑦 ∈ ℝ and 𝑓(0) = 2. Then 𝑓(101) equals __________
Q.43 Let 𝐴 be a 3 × 3 real matrix such that det(𝐴) = 6 and
𝑎𝑑𝑗 𝐴 = (
1 −1 2
5 7 1
−1 1 1
) ,
where 𝑎𝑑𝑗 𝐴 denotes the adjoint of 𝐴.
Then the trace of 𝐴 equals __________ (round off to 2 decimal places)

MS 40/52
Q.44 Let 𝑋 and 𝑌 be two independent and identically distributed random variables
having 𝑈(0, 1) distribution. Then 𝑃(𝑋2
< 𝑌 < 𝑋) equals __________
(round off to 2 decimal places)
Q.45 Consider a sequence of independent Bernoulli trials, where
3
4
is the probability
of success in each trial. Let 𝑋 be a random variable defined as follows: If the
first trial is a success, then 𝑋 counts the number of failures before the next
success. If the first trial is a failure, then 𝑋 counts the number of successes
before the next failure. Then 2𝐸(𝑋) equals __________
Q.46 Let 𝑋 be a random variable denoting the amount of loss in a business. The
moment generating function of 𝑋 is
𝑀(𝑡) = (
2
2 − 𝑡
)
2
, 𝑡 < 2.
If an insurance policy pays 60% of the loss, then the variance of the amount
paid by the insurance policy equals __________ (round off to 2 decimal
places)

MS 41/52
Q.47 Let (𝑋, 𝑌) be a random vector having the joint moment generating function
𝑀(𝑡1, 𝑡2) = (
1
2
𝑒−𝑡1 +
1
2
𝑒𝑡1)
2
(
1
2
+
1
2
𝑒𝑡2)
2
, (𝑡1, 𝑡2) ∈ ℝ2
.
Then 𝑃( |𝑋 + 𝑌| = 2) equals __________ (round off to 2 decimal places)
Q.48 Let 𝑋1 and 𝑋2 be two independent and identically distributed random
variables having 𝜒2
2
distribution and 𝑊 = 𝑋1 + 𝑋2. Then 𝑃(𝑊 > 𝐸(𝑊))
equals __________ (round off to 2 decimal places)
Q.49 Let 2.5, −1.0, 0.5, 1.5 be the observed values of a random sample of size 4
from a continuous distribution with the probability density function
𝑓(𝑥) =
1
8
𝑒−|𝑥−2|
+
3
4√2𝜋
𝑒−
1
2
(𝑥−𝜃)2
, 𝑥 ∈ ℝ,
where 𝜃 ∈ ℝ is unknown. Then the method of moments estimate of 𝜃

MS 42/52
Q.50 Let 𝑋1, 𝑋2, … , 𝑋25 be a random sample from a 𝑁(𝜇, 1) distribution, where
𝜇 ∈ ℝ is unknown. Consider testing of the hypothesis 𝐻0: 𝜇 = 5.2 against
𝐻1: 𝜇 = 5.6. The null hypothesis is rejected if and only if
1
25
∑ 𝑋𝑖 > 𝑘
25
𝑖=1 , for
some constant 𝑘. If the size of the test is 0.05, then the probability of type-II
error equals __________ (round off to 2 decimal places)
Section C: Q.51 – Q.60 Carry TWO marks each.
Q.51 Let 𝑓 ∶ ℝ2
→ ℝ be the function defined by 𝑓(𝑥, 𝑦) = 𝑥2
− 12𝑦.
If 𝑀 and 𝑚 be the maximum value and the minimum value, respectively,
of the function 𝑓 on the circle 𝑥2
+ 𝑦2
= 49, then |𝑀| + |𝑚|
equals __________
Q.52 The value of
∫ ∫ (𝑥 + 𝑦)2
𝑒
2𝑦
𝑥+𝑦 𝑑𝑦 𝑑𝑥
2−𝑥
0
2
0

MS 43/52
Q.53
Let 𝐴 = (
1 −1 2
−1 0 1
2 1 1
) and let (
𝑥1
𝑥2
𝑥3
) be an eigenvector corresponding to
the smallest eigenvalue of 𝐴, satisfying 𝑥1
2
+ 𝑥2
2
+ 𝑥3
2
= 1. Then the value of
|𝑥1| + |𝑥2| + |𝑥3| equals __________ (round off to 2 decimal places)
Q.54 Five men go to a restaurant together and each of them orders a dish that is
different from the dishes ordered by the other members of the group. However,
the waiter serves the dishes randomly. Then the probability that exactly one of
them gets the dish he ordered equals __________ (round off to 2 decimal
places)
Q.55 Let 𝑋 be a random variable having the probability density function
𝑓(𝑥) = {
𝑎𝑥2
+ 𝑏, 0 ≤ 𝑥 ≤ 3
0, otherwise,
where 𝑎 and 𝑏 are real constants, and 𝑃(𝑋 ≥ 2) =
2
3
.
Then 𝐸(𝑋) equals __________ (round off to 2 decimal places)

MS 44/52
Q.56 A vaccine, when it is administered to an individual, produces no side effects
with probability
4
5
, mild side effects with probability
2
15
and severe side effects
with probability
1
15
. Assume that the development of side effects is independent
across individuals. The vaccine was administered to 1000 randomly selected
individuals. If 𝑋1 denotes the number of individuals who developed mild side
effects and 𝑋2 denotes the number of individuals who developed severe side
effects, then the coefficient of variation of 𝑋1 + 𝑋2 equals __________
(round off to 2 decimal places)
Q.57 Let {𝑋𝑛}𝑛≥1 be a sequence of independent and identically distributed random
variables having 𝑈(0, 1) distribution. Let 𝑌𝑛 = 𝑛 min{𝑋1, 𝑋2, … , 𝑋𝑛},
𝑛 ≥ 1. If 𝑌𝑛 converges to 𝑌 in distribution, then the median of 𝑌
Q.58 Let 𝑋(1) < 𝑋(2) < 𝑋(3) < 𝑋(4) < 𝑋(5) be the order statistics based on a
random sample of size 5 from a continuous distribution with the probability
density function
𝑓(𝑥) = {
1
𝑥2
, 1 < 𝑥 < ∞
0, otherwise.
Then the sum of all possible values of 𝑟 ∈ {1, 2, 3, 4, 5} for which 𝐸(𝑋(𝑟))
is finite equals __________

MS 45/52
Q.59 Consider the linear regression model 𝑦𝑖 = 𝛽0 + 𝛽1𝑥𝑖 + 𝜖𝑖, 𝑖 = 1, 2, … , 6,
where 𝛽0 and 𝛽1 are unknown parameters and 𝜖𝑖’s are independent and
identically distributed random variables having 𝑁(0, 1) distribution. The data
on (𝑥𝑖, 𝑦𝑖) are given in the following table.
𝑥𝑖 1.0 2.0 2.5 3.0 3.5 4.5
𝑦𝑖 2.0 3.0 3.5 4.2 5.0 5.4
If 𝛽0
̂ and 𝛽1
̂ are the least squares estimates of 𝛽0 and 𝛽1, respectively,
based on the above data, then 𝛽0
̂ + 𝛽1
̂ equals __________ (round off to 2
decimal places)
Q.60 Let 𝑋1, 𝑋2, … , 𝑋9 be a random sample from a 𝑁(𝜇, 𝜎2
) distribution, where
𝜇 ∈ ℝ and 𝜎 > 0 are unknown. Let the observed values of 𝑋 =
1
9
∑ 𝑋𝑖
9
𝑖=1
and 𝑆2
=
1
8
∑ (𝑋𝑖 − 𝑋)
2
9
𝑖=1 be 9.8 and 1.44, respectively. If the likelihood
ratio test is used to test the hypothesis 𝐻0: 𝜇 = 8.8 against 𝐻1: 𝜇 > 8.8, then
the 𝑝-value of the test equals __________ (round off to 3 decimal places)

IIT JAM Mathematical Statistics - MS 2022 | Sourav Sir's Classes

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