IIT JAM Mathematical Statistics - MS 2022 | Sourav Sir's Classes
1. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 2/52
Special Instructions / Useful Data
Special Instructions / Useful Data
β The set of real numbers
βπ {(π₯1, π₯2, β¦ , π₯π) βΆ π₯π β β, π = 1, 2, β¦ , π}, π = 2, 3, β¦
ln π₯ Natural logarithm of π₯, π₯ > 0
det(π) Determinant of a square matrix π
πΌπ π Γ π identity matrix, π = 2, 3, 4, β¦
πΈπ Complement of a set πΈ
π(πΈ) Probability of an event πΈ
π(πΈ | πΉ) Conditional probability of an event πΈ given the occurrence of the event πΉ
πΈ(π) Expectation of a random variable π
πππ(π) Variance of a random variable π
π(π, π) Continuous uniform distribution on the interval (π, π), ββ < π < π < β
πΈπ₯π(π) Exponential distribution with the probability density function, for π > 0,
π(π₯) = {
π πβ ππ₯
, π₯ > 0
0, otherwise
π(π, π2
) Normal distribution with mean π and variance π2
, π β β, π > 0
Ξ¦(β ) The cumulative distribution function of π(0, 1) distributed random variable
ππ
2 Central chi-square distribution with π degrees of freedom, π = 1, 2, β¦
πΉ
π,π Snedecorβs central πΉ-distribution with (π, π) degrees of freedom,
π, π = 1, 2, β¦
π‘π,πΌ A constant such that π(π > π‘π,πΌ) = πΌ, where π has central Studentβs
π‘-distribution with π degrees of freedom, π = 1, 2, β¦ ; πΌ β (0, 1)
Ξ¦(1.645) = 0.95, Ξ¦(0.355) = 0.6387
π‘8,0.0185 = 2.5
2. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 3/52
Section A: Q.1 β Q.10 Carry ONE mark each.
Q.1 Let {ππ}πβ₯1 be a sequence of non-zero real numbers. Then which one of the
following statements is true?
(A)
If {
ππ+1
ππ
}
πβ₯1
is a convergent sequence, then {ππ}πβ₯1 is also a convergent
sequence
(B) If {ππ}πβ₯1 is a bounded sequence, then {ππ}πβ₯1 is a convergent sequence
(C) If |ππ+2 β ππ+1| β€
3
4
|ππ+1 β ππ| for all π β₯ 1, then {ππ}πβ₯1 is a Cauchy
sequence
(D) If {|ππ|}πβ₯1 is a Cauchy sequence, then {ππ}πβ₯1 is also a Cauchy sequence
3. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 4/52
Q.2 Let π: β β β be the function defined by
π(π₯) = {lim
ββ0
(π₯ + β) sin (
1
π₯
+ β) β π₯ sin
1
π₯
β
, π₯ β 0
0, π₯ = 0.
Then which one of the following statements is NOT true?
(A) π (
2
π
) = 1
(B) π (
1
π
) =
1
π
(C) π (β
2
π
) = β1
(D) π is not continuous at π₯ = 0
4. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 5/52
Q.3 Let π: β β β be the function defined by
π(π₯) = det (
1 + π₯ 9 9
9 1 + π₯ 9
9 9 1 + π₯
) .
Then the maximum value of π on the interval [9, 10] equals
(A) 118
(B) 112
(C) 114
(D) 116
Q.4 Let π΄ and π΅ be two events such that 0 < π(π΄) < 1 and 0 < π(π΅) < 1.
Then which one of the following statements is NOT true?
(A) If π(π΄|π΅) > π(π΄), then π(π΅|π΄) > π(π΅)
(B) If π(π΄ βͺ π΅) = 1, then π΄ and π΅ cannot be independent
(C) If π(π΄|π΅) > π(π΄), then π(π΄π|π΅) < π(π΄π)
(D) If π(π΄|π΅) > π(π΄), then π(π΄π|π΅π) < π(π΄π)
5. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 6/52
Q.5 If π(π‘), π‘ β β, 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) 5πβ5π‘
1β4π‘2
π(π‘), |π‘| <
1
2
(B) πβπ‘
π(π‘), π‘ β β
(C) 1+ππ‘
2(2βππ‘)
π(π‘), π‘ < ln 2
(D) π(4π‘), π‘ β β
Q.6 Let π be a random variable having binomial distribution with parameters
π (> 1) and π (0 < π < 1). Then πΈ (
1
1+π
) equals
(A) 1β(1βπ)π+1
(π+1)π
(B) 1βππ+1
(π+1)(1βπ)
(C) (1βπ)π+1
π(1βπ)
(D) 1βππ
(π+1)π
6. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 7/52
Q.7 Let (π, π) be a random vector having the joint probability density function
π(π₯, π¦) = {
β2
βπ
πβ2π₯
πβ
(π¦βπ₯)2
2 , 0 < π₯ < β, ββ < π¦ < β
0, otherwise.
Then πΈ(π) equals
(A) 1
2
(B) 2
(C) 1
(D) 1
4
7. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 8/52
Q.8 Let π1 and π2 be two independent and identically distributed discrete
random variables having the probability mass function
π(π₯) = {(
1
2
)
π₯
, π₯ = 1, 2, 3, β¦
0, otherwise.
Then π(min{π1, π2} β₯ 5) equals
(A) 1
256
(B) 1
512
(C) 1
64
(D) 9
256
8. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 9/52
Q.9 Let π1, π2, β¦ , ππ (π β₯ 2) be a random sample from πΈπ₯π (
1
π
) distribution,
where π > 0 is unknown. If π =
1
π
β ππ
π
π=1 , then which one of the following
statements is NOT true?
(A) π is the uniformly minimum variance unbiased estimator of π
(B)
π
2
is the uniformly minimum variance unbiased estimator of π2
(C) π
π+1
π
2
is the uniformly minimum variance unbiased estimator of π2
(D) πππ (πΈ(ππ | π )) β€ πππ(ππ)
9. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 10/52
Q.10 Let π1, π2, β¦ , ππ (π β₯ 3) be a random sample from a π(π, π2
)
distribution, where π β β and π > 0 are both unknown. Then which one of
the following is a simple null hypothesis?
(A) π»0: π < 5, π2
= 3
(B) π»0: π = 5, π2
> 3
(C) π»0: π = 5, π2
= 3
(D) π»0: π = 5
Section A: Q.11 β Q.30 Carry TWO marks each.
Q.11
lim
πββ
6
π+2
{(2 +
1
π
)
2
+ (2 +
2
π
)
2
+ β― + (2 +
πβ1
π
)
2
} equals
(A) 38
(B) 36
(C) 32
(D) 30
10. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 11/52
Q.12 Let π: β2
β β be the function defined by
π(π₯, π¦) = { π₯2
sin
1
π₯
+ π¦2
cos π¦ , π₯ β 0
0, π₯ = 0.
Then which one of the following statements is NOT true?
(A) π is continuous at (0, 0)
(B) The partial derivative of π with respect to π₯ is not continuous at (0, 0)
(C) The partial derivative of π with respect to π¦ is continuous at (0, 0)
(D) π is not differentiable at (0, 0)
11. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 12/52
Q.13 Let π: [1, 2] β β be the function defined by
π(π‘) = β« βπ₯2ππ₯2
β 1
π‘
1
ππ₯.
Then the arc length of the graph of π over the interval [1, 2] equals
(A) π2
β βπ
(B) π β βπ
(C) π2
β π
(D) π2
β 1
12. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 13/52
Q.14 Let πΉ: [0, 2] β β be the function defined by
πΉ(π₯) = β« ππ₯ [π‘]
ππ‘,
π₯+2
π₯2
where [π‘] denotes the greatest integer less than or equal to π‘. Then the value
of the derivative of πΉ at π₯ = 1 equals
(A) π3
+ 2π2
β π
(B) π3
β π2
+ 2π
(C) π3
β 2π2
+ π
(D) π3
+ 2π2
+ π
13. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 14/52
Q.15 Let the system of equations
π₯ + ππ¦ + π§ = 1
2π₯ + 4π¦ + π§ = β π
3π₯ + π¦ + 2π§ = π + 2
have infinitely many solutions, where π and π are real constants. Then the
value of 2π + 8π equals
(A) β11
(B) β10
(C) β13
(D) β14
14. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 15/52
Q.16
Let π΄ = (
0 1 0
1 0 0
0 1 1
). Then the sum of all the elements of π΄100
equals
(A) 101
(B) 103
(C) 102
(D) 100
15. JAM 2022 MATHEMATICAL STATISTICS - MS
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
16. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 17/52
Q.18 A year is chosen at random from the set of years {2012, 2013, β¦ , 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
17. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 18/52
Q.19 Suppose that a fair coin is tossed repeatedly and independently. Let π denote
the number of tosses required to obtain for the first time a tail that is
immediately preceded by a head. Then πΈ(π) and π(π > 4), respectively,
are
(A) 4 and
5
16
(B) 4 and
11
16
(C) 6 and
5
16
(D) 6 and
11
16
18. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 19/52
Q.20 Let π be a random variable with the moment generating function
π(π‘) =
1
(1 β 4π‘)5
, π‘ <
1
4
.
Then the lower bounds for π(π < 40), using Chebyshevβs inequality and
Markovβs 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
19. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 20/52
Q.21 In a store, the daily demand for milk (in litres) is a random variable having
πΈπ₯π(π) distribution, where π > 0. At the beginning of the day, the store
purchases π (> 0) litres of milk at a fixed price π (> 0) per litre. The milk
is then sold to the customers at a fixed price π (> π) per litre. At the end of
the day, the unsold milk is discarded. Then the value of π that maximizes the
expected net profit for the store equals
(A)
β
1
π
ln (
π
π
)
(B)
β
1
π
ln (
π
π +π
)
(C)
β
1
π
ln (
π βπ
π
)
(D)
β
1
π
ln (
π
π +π
)
20. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 21/52
Q.22 Let π1, π2 and π3 be three independent and identically distributed random
variables having π(0, 1) distribution. Then πΈ [(
ln π1
ln π1π2π3
)
2
] equals
(A) 1
6
(B) 1
3
(C) 1
8
(D) 1
4
21. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 22/52
Q.23 Let (π, π) be a random vector having bivariate normal distribution with
parameters πΈ(π) = 0, πππ(π) = 1, πΈ(π) = β1, πππ(π) = 4 and
π(π, π) = β
1
2
, where π(π, π) denotes the correlation coefficient between
π and π. Then π(π + π > 1 | 2π β π = 1) equals
(A) Ξ¦ (β
1
2
)
(B) Ξ¦ (β
1
3
)
(C) Ξ¦ (β
1
4
)
(D) Ξ¦ (β
4
3
)
22. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 23/52
Q.24 Let {ππ}πβ₯1 be a sequence of independent and identically distributed random
variables having the common probability density function
π(π₯) = {
2
π₯3
, π₯ β₯ 1
0, otherwise.
If lim
πββ
π (|
1
π
β ππ
π
π=1 β π| < π) = 1 for all π > 0, then π equals
(A) 4
(B) 2
(C) ln 4
(D) ln 2
23. JAM 2022 MATHEMATICAL STATISTICS - MS
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
π(π₯) =
{
1, 0 < π₯ β€
1
2
1
2π β 1
,
1
2
< π₯ β€ π
0, otherwise,
where π >
1
2
is unknown. Then the maximum likelihood estimate and the
method of moments estimate of π, 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
24. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 25/52
Q.26 For π = 1, 2, 3, β¦, let the joint moment generating function of (π, ππ) be
ππ,ππ
(π‘1, π‘2) = π
π‘1
2
2 (1 β 2π‘2)β
π
2 , π‘1 β β, π‘2 <
1
2
.
If ππ =
βπ π
βππ
, π β₯ 1, then which one of the following statements is true?
(A) The minimum value of π for which πππ(ππ) is finite is 2
(B) πΈ(π10
3 ) = 10
(C) πππ(π + π4) = 7
(D)
lim
πββ
π(|ππ| > 3) = 1 β
β2
βπ
β« πβ
π‘2
2 ππ‘
3
0
25. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 26/52
Q.27 Let π(1) < π(2) < β― < π(9) be the order statistics corresponding to a
random sample of size 9 from π(0, 1) distribution. Then which one of the
following statements is NOT true?
(A)
πΈ (
π(9)
1βπ(9)
) is finite
(B) πΈ(π(5)) = 0.5
(C) The median of π(5) is 0.5
(D) The mode of π(5) is 0.5
26. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 27/52
Q.28 Let π1, π2, β¦ , π16 be a random sample from a π(4π, 1) distribution and
π1, π2, β¦ , π8 be a random sample from a π(π, 1) distribution, where
π β β is unknown. Assume that the two random samples are independent.
If you are looking for a confidence interval for π based on the statistic
8π + π, where π =
1
16
β ππ
16
π=1 and π =
1
8
β ππ
8
π=1 , then which one of the
following statements is true?
(A) There exists a 90% confidence interval for π of length less than 0.1
(B) There exists a 90% confidence interval for π of length greater than 0.3
(C)
[
8π+π
33
β
1.645
2β66
,
8π+π
33
+
1.645
2β66
] is the unique 90% confidence interval for π
(D) π always belongs to its 90% confidence interval
27. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 28/52
Q.29 Let π1, π2, π3, π4 be a random sample from a distribution with the probability
mass function
π(π₯) = {
ππ₯(1 β π)1βπ₯
, π₯ = 0, 1
0, otherwise,
where π β (0, 1) is unknown. Let 0 < πΌ β€ 1. To test the hypothesis
π»0: π =
1
2
against π»1: π >
1
2
, consider the size πΌ test that rejects π»0 if and
only if β ππ
4
π=1 β₯ ππΌ, for some ππΌ β {0, 1, 2, 3, 4}. Then for which one of
the following values of πΌ, the size πΌ test does NOT exist?
(A) 1
16
(B) 1
4
(C) 11
16
(D) 5
16
28. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 29/52
Q.30 Let π1, π2, π3, π4 be a random sample from a Poisson distribution with
unknown mean π > 0. For testing the hypothesis
π»0: π = 1 against π»1: π = 1.5,
let π½ denote the power of the test that rejects π»0 if and only if β ππ
4
π=1 β₯ 5.
Then which one of the following statements is true?
(A) π½ > 0.80
(B) 0.75 < π½ β€ 0.80
(C) 0.70 < π½ β€ 0.75
(D) 0.65 < π½ β€ 0.70
29. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 30/52
Section B: Q.31 β Q.40 Carry TWO marks each.
Q.31 Let {ππ}πβ₯1 be a sequence of real numbers such that ππ =
1
3π
for all π β₯ 1.
Then which of the following statements is/are true?
(A) β (β1)π+1
ππ
β
π=1 is a convergent series
(B) β
(β1)π+1
π
β
π=1 (π1 + π2 + β¦ + ππ) is a convergent series
(C)
The radius of convergence of the power series β πππ₯π
β
π=1 is
1
3
(D) β ππ sin
1
ππ
β
π=1 is a convergent series
30. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 31/52
Q.32 Let π: β2
β β be the function defined by
π(π₯, π¦) = 8(π₯2
β π¦2) β π₯4
+ π¦4
.
Then which of the following statements is/are true?
(A) π has 9 critical points
(B) π has a saddle point at (2, 2)
(C) π has a local maximum at (β2, 0)
(D) π has a local minimum at (0, β2)
Q.33 If π β₯ 2, then which of the following statements is/are true?
(A) If π΄ and π΅ are π Γ π real orthogonal matrices such that
det(π΄) + det(π΅) = 0, then π΄ + π΅ is a singular matrix
(B) If π΄ is an π Γ π real matrix such that πΌπ + π΄ is non-singular, then
πΌπ + (πΌπ + π΄)β1
(πΌπ β π΄) is a singular matrix
(C) If π΄ is an π Γ π real skew-symmetric matrix, then πΌπ β π΄2
is a non-
singular matrix
(D) If π΄ is an π Γ π real orthogonal matrix, then det(π΄ β ππΌπ) β 0 for all
π β {π₯ β β βΆ π₯ β Β±1}
32. JAM 2022 MATHEMATICAL STATISTICS - MS
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 π(250, 1750) distribution and π denotes the amount
the employee gets from the insurance policy, then which of the following
statements is/are true?
(A) πΈ(π) =
500
3
(B) π(π > 300) =
3
10
(C) The median of π is zero
(D) The quantile of order 0.6 for π equals 100
33. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 34/52
Q.36 Let π and π be two independent random variables having π(0, π1
2
) and
π(0, π2
2
) distributions, respectively, where 0 < π1 < π2. Then which of the
following statements is/are true?
(A) π + π and π β π are independent
(B) 2π + π and π β π are independent if 2π1
2
= π2
2
(C) π + π and π β π are identically distributed
(D) π + π and 2π β π are independent if 2π1
2
= π2
2
34. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 35/52
Q.37 Let (π, π) be a discrete random vector. Then which of the following
statements is/are true?
(A) If π and π are independent, then π2
and |π| are also independent.
(B) If the correlation coefficient between π and π is 1, then
π(π = ππ + π) = 1 for some π, π β β
(C) If π and π are independent and πΈ[(ππ)2] = 0, then π(π = 0) = 1 or
π(π = 0) = 1
(D) If πππ(π) = 0, then π and π are independent
35. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 36/52
Q.38 Let π1, π2 and π3 be three independent and identically distributed random
variables having π(0, 1) distribution. If
π =
2π1
2
(π2 + π3)2
and π =
2(π2 β π3)2
2π1
2
+ (π2 + π3)2
,
then which of the following statements is/are true?
(A) π has πΉ1,1 distribution and π has πΉ1,2 distribution
(B) π has πΉ1,1 distribution and π has πΉ2,1 distribution
(C) π and π are independent
(D) 1
2
π(1 + π) has πΉ2,3 distribution
36. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 37/52
Q.39 Let π1, π2, π3, π4 be a random sample from a continuous distribution with
the probability density function π(π₯) =
1
2
πβ|π₯βπ|
, π₯ β β, where π β β is
unknown. Let the corresponding order statistics be denoted by
π(1) < π(2) < π(3) < π(4). Then which of the following statements is/are
true?
(A) 1
2
(π(2) + π(3)) is the unique maximum likelihood estimator of π
(B) (π(1), π(2), π(3), π(4)) is a sufficient statistic for π
(C) 1
4
(π(2) + π(3)) (π(2) + π(3) + 2) is a maximum likelihood estimator of
π(π + 1)
(D) (π1π2π3, π1π2π4) is a complete statistic
37. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 38/52
Q.40 Let π1, π2, β¦ , ππ (π > 1) be a random sample from a π(π, 1) distribution,
where π β β is unknown. Let 0 < πΌ < 1. To test the hypothesis π»0: π = 0
against π»1: π = πΏ, where πΏ > 0 is a constant, let π½ denote the power of the
size πΌ test that rejects π»0 if and only if
1
π
β ππ > ππΌ
π
π=1 , for some constant
ππΌ. Then which of the following statements is/are true?
(A) For a fixed value of πΏ, π½ increases as πΌ increases
(B) For a fixed value of πΌ, π½ increases as πΏ increases
(C) For a fixed value of πΏ, π½ decreases as πΌ increases
(D) For a fixed value of πΌ, π½ decreases as πΏ increases
38. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 39/52
Section C: Q.41 β Q.50 Carry ONE mark each.
Q.41 Let {ππ}πβ₯1 be a sequence of real numbers such that π1+5π = 2,
π2+5π = 3, π3+5π = 4, π4+5π = 5, π5+5π = 6, π = 0, 1, 2, β¦ . Then
limsupnββ ππ + liminfnββ ππ equals __________
Q.42 Let π: β β β be a function such that
20(π₯ β π¦) β€ π(π₯) β π(π¦) β€ 20(π₯ β π¦) + 2(π₯ β π¦)2
for all π₯, π¦ β β and π(0) = 2. Then π(101) equals __________
Q.43 Let π΄ be a 3 Γ 3 real matrix such that det(π΄) = 6 and
πππ π΄ = (
1 β1 2
5 7 1
β1 1 1
) ,
where πππ π΄ denotes the adjoint of π΄.
Then the trace of π΄ equals __________ (round off to 2 decimal places)
39. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 40/52
Q.44 Let π and π be two independent and identically distributed random variables
having π(0, 1) distribution. Then π(π2
< π < π) 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 π be a random variable defined as follows: If the
first trial is a success, then π counts the number of failures before the next
success. If the first trial is a failure, then π counts the number of successes
before the next failure. Then 2πΈ(π) equals __________
Q.46 Let π be a random variable denoting the amount of loss in a business. The
moment generating function of π is
π(π‘) = (
2
2 β π‘
)
2
, π‘ < 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)
40. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 41/52
Q.47 Let (π, π) be a random vector having the joint moment generating function
π(π‘1, π‘2) = (
1
2
πβπ‘1 +
1
2
ππ‘1)
2
(
1
2
+
1
2
ππ‘2)
2
, (π‘1, π‘2) β β2
.
Then π( |π + π| = 2) equals __________ (round off to 2 decimal places)
Q.48 Let π1 and π2 be two independent and identically distributed random
variables having π2
2
distribution and π = π1 + π2. Then π(π > πΈ(π))
equals __________ (round off to 2 decimal places)
Q.49 Let 2.5, β1.0, 0.5, 1.5 be the observed values of a random sample of size 4
from a continuous distribution with the probability density function
π(π₯) =
1
8
πβ|π₯β2|
+
3
4β2π
πβ
1
2
(π₯βπ)2
, π₯ β β,
where π β β is unknown. Then the method of moments estimate of π
equals __________ (round off to 2 decimal places)
41. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 42/52
Q.50 Let π1, π2, β¦ , π25 be a random sample from a π(π, 1) distribution, where
π β β is unknown. Consider testing of the hypothesis π»0: π = 5.2 against
π»1: π = 5.6. The null hypothesis is rejected if and only if
1
25
β ππ > π
25
π=1 , for
some constant π. 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 β Q.60 Carry TWO marks each.
Q.51 Let π βΆ β2
β β be the function defined by π(π₯, π¦) = π₯2
β 12π¦.
If π and π be the maximum value and the minimum value, respectively,
of the function π on the circle π₯2
+ π¦2
= 49, then |π| + |π|
equals __________
Q.52 The value of
β« β« (π₯ + π¦)2
π
2π¦
π₯+π¦ ππ¦ ππ₯
2βπ₯
0
2
0
equals __________ (round off to 2 decimal places)
42. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 43/52
Q.53
Let π΄ = (
1 β1 2
β1 0 1
2 1 1
) and let (
π₯1
π₯2
π₯3
) be an eigenvector corresponding to
the smallest eigenvalue of π΄, satisfying π₯1
2
+ π₯2
2
+ π₯3
2
= 1. Then the value of
|π₯1| + |π₯2| + |π₯3| 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 π be a random variable having the probability density function
π(π₯) = {
ππ₯2
+ π, 0 β€ π₯ β€ 3
0, otherwise,
where π and π are real constants, and π(π β₯ 2) =
2
3
.
Then πΈ(π) equals __________ (round off to 2 decimal places)
43. JAM 2022 MATHEMATICAL STATISTICS - MS
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 π1 denotes the number of individuals who developed mild side
effects and π2 denotes the number of individuals who developed severe side
effects, then the coefficient of variation of π1 + π2 equals __________
(round off to 2 decimal places)
Q.57 Let {ππ}πβ₯1 be a sequence of independent and identically distributed random
variables having π(0, 1) distribution. Let ππ = π min{π1, π2, β¦ , ππ},
π β₯ 1. If ππ converges to π in distribution, then the median of π
equals __________ (round off to 2 decimal places)
Q.58 Let π(1) < π(2) < π(3) < π(4) < π(5) be the order statistics based on a
random sample of size 5 from a continuous distribution with the probability
density function
π(π₯) = {
1
π₯2
, 1 < π₯ < β
0, otherwise.
Then the sum of all possible values of π β {1, 2, 3, 4, 5} for which πΈ(π(π))
is finite equals __________
44. JAM 2022 MATHEMATICAL STATISTICS - MS
MS 45/52
Q.59 Consider the linear regression model π¦π = π½0 + π½1π₯π + ππ, π = 1, 2, β¦ , 6,
where π½0 and π½1 are unknown parameters and ππβs are independent and
identically distributed random variables having π(0, 1) distribution. The data
on (π₯π, π¦π) are given in the following table.
π₯π 1.0 2.0 2.5 3.0 3.5 4.5
π¦π 2.0 3.0 3.5 4.2 5.0 5.4
If π½0
Μ and π½1
Μ are the least squares estimates of π½0 and π½1, respectively,
based on the above data, then π½0
Μ + π½1
Μ equals __________ (round off to 2
decimal places)
Q.60 Let π1, π2, β¦ , π9 be a random sample from a π(π, π2
) distribution, where
π β β and π > 0 are unknown. Let the observed values of π =
1
9
β ππ
9
π=1
and π2
=
1
8
β (ππ β π)
2
9
π=1 be 9.8 and 1.44, respectively. If the likelihood
ratio test is used to test the hypothesis π»0: π = 8.8 against π»1: π > 8.8, then
the π-value of the test equals __________ (round off to 3 decimal places)