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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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