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Page 1
JAM 2022 MATHEMATICAL STATISTICS - MS
MS 2/52
Special Instructions / Useful Data
Special Instructions / Useful Data
R The set of real numbers
R
?? {(?? 1
,?? 2
,…,?? ?? ):?? ?? ?R,?? =1,2,…,?? }, ?? =2,3,…
ln?? Natural logarithm of ?? , ?? >0
det(?? ) Determinant of a square matrix ??
?? ?? ?? ×?? identity matrix, ?? =2,3,4,…
?? ?? Complement of a set ??
?? (?? ) Probability of an event ??
?? (?? | ?? ) Conditional probability of an event ?? given the occurrence of the event ??
?? (?? ) Expectation of a random variable ??
?????? (?? ) Variance of a random variable ??
?? (?? ,?? ) Continuous uniform distribution on the interval (?? ,?? ), -8<?? <?? <8
?????? (?? ) Exponential distribution with the probability density function, for ?? >0,
?? (?? )={
?? ?? - ????
, ?? >0
0, otherwise
?? (?? ,?? 2
) Normal distribution with mean ?? and variance ?? 2
, ?? ?R, ?? >0
F(·) The cumulative distribution function of ?? (0,1) distributed random variable
?? ?? 2
Central chi-square distribution with ?? degrees of freedom, ?? =1,2,…
?? ?? ,??
Snedecor’s central ?? -distribution with (?? ,?? ) degrees of freedom,
?? ,?? =1,2,…
?? ?? ,??
A constant such that ?? (?? >?? ?? ,?? )=?? , where ?? has central Student’s
?? -distribution with ?? degrees of freedom, ?? =1,2,…; ?? ?(0,1)
F(1.645)=0.95, F(0.355)=0.6387
?? 8,0.0185
=2.5
Page 2
JAM 2022 MATHEMATICAL STATISTICS - MS
MS 2/52
Special Instructions / Useful Data
Special Instructions / Useful Data
R The set of real numbers
R
?? {(?? 1
,?? 2
,…,?? ?? ):?? ?? ?R,?? =1,2,…,?? }, ?? =2,3,…
ln?? Natural logarithm of ?? , ?? >0
det(?? ) Determinant of a square matrix ??
?? ?? ?? ×?? identity matrix, ?? =2,3,4,…
?? ?? Complement of a set ??
?? (?? ) Probability of an event ??
?? (?? | ?? ) Conditional probability of an event ?? given the occurrence of the event ??
?? (?? ) Expectation of a random variable ??
?????? (?? ) Variance of a random variable ??
?? (?? ,?? ) Continuous uniform distribution on the interval (?? ,?? ), -8<?? <?? <8
?????? (?? ) Exponential distribution with the probability density function, for ?? >0,
?? (?? )={
?? ?? - ????
, ?? >0
0, otherwise
?? (?? ,?? 2
) Normal distribution with mean ?? and variance ?? 2
, ?? ?R, ?? >0
F(·) The cumulative distribution function of ?? (0,1) distributed random variable
?? ?? 2
Central chi-square distribution with ?? degrees of freedom, ?? =1,2,…
?? ?? ,??
Snedecor’s central ?? -distribution with (?? ,?? ) degrees of freedom,
?? ,?? =1,2,…
?? ?? ,??
A constant such that ?? (?? >?? ?? ,?? )=?? , where ?? has central Student’s
?? -distribution with ?? degrees of freedom, ?? =1,2,…; ?? ?(0,1)
F(1.645)=0.95, F(0.355)=0.6387
?? 8,0.0185
=2.5
JAM 2022 MATHEMATICAL STATISTICS - MS
MS 3/52
Section A: Q.1 – Q.10 Carry ONE mark each.
Q.1 Let {?? ?? }
?? =1
be a sequence of non-zero real numbers. Then which one of the
following statements is true?
(A)
If {
?? ?? +1
?? ?? }
?? =1
is a convergent sequence, then {?? ?? }
?? =1
is also a convergent
sequence
(B) If {?? ?? }
?? =1
is a bounded sequence, then {?? ?? }
?? =1
is a convergent sequence
(C)
If |?? ?? +2
-?? ?? +1
|=
3
4
|?? ?? +1
-?? ?? | for all ?? =1, then {?? ?? }
?? =1
is a Cauchy
sequence
(D) If {|?? ?? |}
?? =1
is a Cauchy sequence, then {?? ?? }
?? =1
is also a Cauchy sequence
Page 3
JAM 2022 MATHEMATICAL STATISTICS - MS
MS 2/52
Special Instructions / Useful Data
Special Instructions / Useful Data
R The set of real numbers
R
?? {(?? 1
,?? 2
,…,?? ?? ):?? ?? ?R,?? =1,2,…,?? }, ?? =2,3,…
ln?? Natural logarithm of ?? , ?? >0
det(?? ) Determinant of a square matrix ??
?? ?? ?? ×?? identity matrix, ?? =2,3,4,…
?? ?? Complement of a set ??
?? (?? ) Probability of an event ??
?? (?? | ?? ) Conditional probability of an event ?? given the occurrence of the event ??
?? (?? ) Expectation of a random variable ??
?????? (?? ) Variance of a random variable ??
?? (?? ,?? ) Continuous uniform distribution on the interval (?? ,?? ), -8<?? <?? <8
?????? (?? ) Exponential distribution with the probability density function, for ?? >0,
?? (?? )={
?? ?? - ????
, ?? >0
0, otherwise
?? (?? ,?? 2
) Normal distribution with mean ?? and variance ?? 2
, ?? ?R, ?? >0
F(·) The cumulative distribution function of ?? (0,1) distributed random variable
?? ?? 2
Central chi-square distribution with ?? degrees of freedom, ?? =1,2,…
?? ?? ,??
Snedecor’s central ?? -distribution with (?? ,?? ) degrees of freedom,
?? ,?? =1,2,…
?? ?? ,??
A constant such that ?? (?? >?? ?? ,?? )=?? , where ?? has central Student’s
?? -distribution with ?? degrees of freedom, ?? =1,2,…; ?? ?(0,1)
F(1.645)=0.95, F(0.355)=0.6387
?? 8,0.0185
=2.5
JAM 2022 MATHEMATICAL STATISTICS - MS
MS 3/52
Section A: Q.1 – Q.10 Carry ONE mark each.
Q.1 Let {?? ?? }
?? =1
be a sequence of non-zero real numbers. Then which one of the
following statements is true?
(A)
If {
?? ?? +1
?? ?? }
?? =1
is a convergent sequence, then {?? ?? }
?? =1
is also a convergent
sequence
(B) If {?? ?? }
?? =1
is a bounded sequence, then {?? ?? }
?? =1
is a convergent sequence
(C)
If |?? ?? +2
-?? ?? +1
|=
3
4
|?? ?? +1
-?? ?? | for all ?? =1, then {?? ?? }
?? =1
is a Cauchy
sequence
(D) If {|?? ?? |}
?? =1
is a Cauchy sequence, then {?? ?? }
?? =1
is also a Cauchy sequence
JAM 2022 MATHEMATICAL STATISTICS - MS
MS 4/52
Q.2
Let ?? :R?R be the function defined by
?? (?? )={
lim
h?0
(?? +h)sin(
1
?? +h)-?? sin
1
?? h
, ?? ?0
0, ?? =0.
Then which one of the following statements is NOT true?
(A)
?? (
2
?? )=1
(B)
?? (
1
?? )=
1
??
(C)
?? (-
2
?? )=-1
(D)
?? is not continuous at ?? =0
Page 4
JAM 2022 MATHEMATICAL STATISTICS - MS
MS 2/52
Special Instructions / Useful Data
Special Instructions / Useful Data
R The set of real numbers
R
?? {(?? 1
,?? 2
,…,?? ?? ):?? ?? ?R,?? =1,2,…,?? }, ?? =2,3,…
ln?? Natural logarithm of ?? , ?? >0
det(?? ) Determinant of a square matrix ??
?? ?? ?? ×?? identity matrix, ?? =2,3,4,…
?? ?? Complement of a set ??
?? (?? ) Probability of an event ??
?? (?? | ?? ) Conditional probability of an event ?? given the occurrence of the event ??
?? (?? ) Expectation of a random variable ??
?????? (?? ) Variance of a random variable ??
?? (?? ,?? ) Continuous uniform distribution on the interval (?? ,?? ), -8<?? <?? <8
?????? (?? ) Exponential distribution with the probability density function, for ?? >0,
?? (?? )={
?? ?? - ????
, ?? >0
0, otherwise
?? (?? ,?? 2
) Normal distribution with mean ?? and variance ?? 2
, ?? ?R, ?? >0
F(·) The cumulative distribution function of ?? (0,1) distributed random variable
?? ?? 2
Central chi-square distribution with ?? degrees of freedom, ?? =1,2,…
?? ?? ,??
Snedecor’s central ?? -distribution with (?? ,?? ) degrees of freedom,
?? ,?? =1,2,…
?? ?? ,??
A constant such that ?? (?? >?? ?? ,?? )=?? , where ?? has central Student’s
?? -distribution with ?? degrees of freedom, ?? =1,2,…; ?? ?(0,1)
F(1.645)=0.95, F(0.355)=0.6387
?? 8,0.0185
=2.5
JAM 2022 MATHEMATICAL STATISTICS - MS
MS 3/52
Section A: Q.1 – Q.10 Carry ONE mark each.
Q.1 Let {?? ?? }
?? =1
be a sequence of non-zero real numbers. Then which one of the
following statements is true?
(A)
If {
?? ?? +1
?? ?? }
?? =1
is a convergent sequence, then {?? ?? }
?? =1
is also a convergent
sequence
(B) If {?? ?? }
?? =1
is a bounded sequence, then {?? ?? }
?? =1
is a convergent sequence
(C)
If |?? ?? +2
-?? ?? +1
|=
3
4
|?? ?? +1
-?? ?? | for all ?? =1, then {?? ?? }
?? =1
is a Cauchy
sequence
(D) If {|?? ?? |}
?? =1
is a Cauchy sequence, then {?? ?? }
?? =1
is also a Cauchy sequence
JAM 2022 MATHEMATICAL STATISTICS - MS
MS 4/52
Q.2
Let ?? :R?R be the function defined by
?? (?? )={
lim
h?0
(?? +h)sin(
1
?? +h)-?? sin
1
?? h
, ?? ?0
0, ?? =0.
Then which one of the following statements is NOT true?
(A)
?? (
2
?? )=1
(B)
?? (
1
?? )=
1
??
(C)
?? (-
2
?? )=-1
(D)
?? is not continuous at ?? =0
JAM 2022 MATHEMATICAL STATISTICS - MS
MS 5/52
Q.3 Let ?? :R?R be the function defined by
?? (?? )=det(
1+?? 9 9
9 1+?? 9
9 9 1+?? ).
Then the maximum value of ?? on the interval [9,10] equals
(A)
118
(B)
112
(C)
114
(D)
116
Q.4
Let ?? and ?? be two events such that 0<?? (?? )<1 and 0<?? (?? )<1.
Then which one of the following statements is NOT true?
(A)
If ?? (?? |?? )>?? (?? ), then ?? (?? |?? )>?? (?? )
(B)
If ?? (?? ??? )=1, then ?? and ?? cannot be independent
(C)
If ?? (?? |?? )>?? (?? ), then ?? (?? ?? |?? )<?? (?? ?? )
(D)
If ?? (?? |?? )>?? (?? ), then ?? (?? ?? |?? ?? )<?? (?? ?? )
Page 5
JAM 2022 MATHEMATICAL STATISTICS - MS
MS 2/52
Special Instructions / Useful Data
Special Instructions / Useful Data
R The set of real numbers
R
?? {(?? 1
,?? 2
,…,?? ?? ):?? ?? ?R,?? =1,2,…,?? }, ?? =2,3,…
ln?? Natural logarithm of ?? , ?? >0
det(?? ) Determinant of a square matrix ??
?? ?? ?? ×?? identity matrix, ?? =2,3,4,…
?? ?? Complement of a set ??
?? (?? ) Probability of an event ??
?? (?? | ?? ) Conditional probability of an event ?? given the occurrence of the event ??
?? (?? ) Expectation of a random variable ??
?????? (?? ) Variance of a random variable ??
?? (?? ,?? ) Continuous uniform distribution on the interval (?? ,?? ), -8<?? <?? <8
?????? (?? ) Exponential distribution with the probability density function, for ?? >0,
?? (?? )={
?? ?? - ????
, ?? >0
0, otherwise
?? (?? ,?? 2
) Normal distribution with mean ?? and variance ?? 2
, ?? ?R, ?? >0
F(·) The cumulative distribution function of ?? (0,1) distributed random variable
?? ?? 2
Central chi-square distribution with ?? degrees of freedom, ?? =1,2,…
?? ?? ,??
Snedecor’s central ?? -distribution with (?? ,?? ) degrees of freedom,
?? ,?? =1,2,…
?? ?? ,??
A constant such that ?? (?? >?? ?? ,?? )=?? , where ?? has central Student’s
?? -distribution with ?? degrees of freedom, ?? =1,2,…; ?? ?(0,1)
F(1.645)=0.95, F(0.355)=0.6387
?? 8,0.0185
=2.5
JAM 2022 MATHEMATICAL STATISTICS - MS
MS 3/52
Section A: Q.1 – Q.10 Carry ONE mark each.
Q.1 Let {?? ?? }
?? =1
be a sequence of non-zero real numbers. Then which one of the
following statements is true?
(A)
If {
?? ?? +1
?? ?? }
?? =1
is a convergent sequence, then {?? ?? }
?? =1
is also a convergent
sequence
(B) If {?? ?? }
?? =1
is a bounded sequence, then {?? ?? }
?? =1
is a convergent sequence
(C)
If |?? ?? +2
-?? ?? +1
|=
3
4
|?? ?? +1
-?? ?? | for all ?? =1, then {?? ?? }
?? =1
is a Cauchy
sequence
(D) If {|?? ?? |}
?? =1
is a Cauchy sequence, then {?? ?? }
?? =1
is also a Cauchy sequence
JAM 2022 MATHEMATICAL STATISTICS - MS
MS 4/52
Q.2
Let ?? :R?R be the function defined by
?? (?? )={
lim
h?0
(?? +h)sin(
1
?? +h)-?? sin
1
?? h
, ?? ?0
0, ?? =0.
Then which one of the following statements is NOT true?
(A)
?? (
2
?? )=1
(B)
?? (
1
?? )=
1
??
(C)
?? (-
2
?? )=-1
(D)
?? is not continuous at ?? =0
JAM 2022 MATHEMATICAL STATISTICS - MS
MS 5/52
Q.3 Let ?? :R?R be the function defined by
?? (?? )=det(
1+?? 9 9
9 1+?? 9
9 9 1+?? ).
Then the maximum value of ?? on the interval [9,10] equals
(A)
118
(B)
112
(C)
114
(D)
116
Q.4
Let ?? and ?? be two events such that 0<?? (?? )<1 and 0<?? (?? )<1.
Then which one of the following statements is NOT true?
(A)
If ?? (?? |?? )>?? (?? ), then ?? (?? |?? )>?? (?? )
(B)
If ?? (?? ??? )=1, then ?? and ?? cannot be independent
(C)
If ?? (?? |?? )>?? (?? ), then ?? (?? ?? |?? )<?? (?? ?? )
(D)
If ?? (?? |?? )>?? (?? ), then ?? (?? ?? |?? ?? )<?? (?? ?? )
JAM 2022 MATHEMATICAL STATISTICS - MS
MS 6/52
Q.5
If ?? (?? ), ?? ?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)
5?? -5?? 1-4?? 2
?? (?? ), |?? |<
1
2
(B)
?? -?? ?? (?? ), ?? ?R
(C)
1+?? ?? 2(2-?? ?? )
?? (?? ), ?? <ln2
(D)
?? (4?? ), ?? ?R
Q.6
Let ?? be a random variable having binomial distribution with parameters
?? (>1) and ?? (0<?? <1). Then ?? (
1
1+?? ) equals
(A)
1-(1-?? )
?? +1
(?? +1)??
(B)
1-?? ?? +1
(?? +1)(1-?? )
(C)
(1-?? )
?? +1
?? (1-?? )
(D)
1-?? ??
(?? +1)??
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Nov 22, 2024 Last updated |
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