IIT JAM 2016 Question Paper Mathematical Statistics (MS)

JAM 2016 MATHEMATICAL STATISTICS - MS

Special Instructions / Useful Data
 Set of all real numbers

n  x ,, x  : x  , i  1,, n
1 n i

P  A Probability of an event A

i.i.d. Independently and identically distributed

Bin  n, p  Binomial distribution with parameters n and p

Poisson   Poisson distribution with mean 

N   , 2  Normal distribution with mean  and variance  2

The exponential distribution with probability density function

 e   x , x  0,
Exp    f x |    ,  0
0, otherwise

tn Student’s t distribution with n degrees of freedom

 n2 Chi-square distribution with n degrees of freedom

 n2,  
A constant such that P W   n2,   , where W has  n2 distribution

  x Cumulative distribution function of N  0,1

  x Probability density function of N  0,1

AC Complement of an event A

EX  Expectation of a random variable X

Var  X  Variance of a random variable X
1

B  m, n   x 1  x 
m 1 n 1
dx, m  0, n  0
0

 x The greatest integer less than or equal to real number x

f Derivative of function f

  0.25  0.5987,   0.5  0.6915,   0.625   0.7341,   0.71  0.7612,

 1  0.8413,  1.125   0.8697,   2   0.9772

MS 2/18

JAM 2016 MATHEMATICAL STATISTICS - MS

SECTION – A
MULTIPLE CHOICE QUESTIONS (MCQ)
Q. 1 – Q.10 carry one mark each.

Q.1 Let
1 2 0 2
 1 2 1 1 
P .
 1 2 3 7 
 
 1 2 2 4 
Then rank of P equals
(A) 4
(B) 3
(C) 2
(D) 1

Q.2 Let  ,  ,  be real numbers such that   0 and   0. Suppose

 
P
0 
,

and P 1  P. Then
(A)   0 and    1
(B)   0 and    1
(C)   0 and    2
(D)   0 and    1

Q.3 Let m  1. The volume of the solid generated by revolving the region between the y-axis and the
curve x y  4, 1  y  m, about the y-axis is 15  . The value of m is

(A) 14 (B) 15 (C) 16 (D) 17

Q.4 Consider the region S enclosed by the surface z  y 2 and the planes z  1, x  0, x  1, y  1
and y  1. The volume of S is

1 2 (C) 1
4
(A) (B) (D)
3 3 3

MS 3/18

JAM 2016 MATHEMATICAL STATISTICS - MS

Q.5 Let X be a discrete random variable with the moment generating function

M X t   e
  , t  .
0.5 e t 1

Then P  X  1 equals

3 1 2 1 1 2   e 1 2
(A) e  1 2 (B) e (C) e (D) e
2 2

Q.6 Let E and F be two independent events with
2
P  E | F   P  F | E   1, P  E  F   and P  F   P  E  .
9
Then P  E  equals

1 1 2 3
(A) (B) (C) (D)
3 2 3 4

Q.7 Let X be a continuous random variable with the probability density function

1
f ( x)  , x  .
(2  x 2 )3/2

Then E ( X 2 )

(A) equals 0 (B) equals 1
(C) equals 2 (D) does not exist

Q.8 The probability density function of a random variable X is given by

 x  1 , 0  x 1
f ( x)   ,   0.
0, otherwise

Then the distribution of the random variable Y  log e X  2  is

1 2
(A)  22 (B) 2 (C) 2  22 (D) 12
2

Q.9 1 n 2
Let X 1 , X 2 , be a sequence of i.i.d. N (0,1) random variables. Then, as n  ,  Xi
n i 1
converges in probability to
(A) 0 (B) 0.5 (C) 1 (D) 2

MS 4/18

JAM 2016 MATHEMATICAL STATISTICS - MS

Q.10 Consider the simple linear regression model with n random observations Yi   0  1 xi   i ,
i  1, , n,  n  2  .  0 and 1 are unknown parameters, x1 , , xn are observed values of the
regressor variable and 1 , ,  n are error random variables with E   i   0, i  1, , n, and for
 0, if i  j ,
i, j  1, , n, Cov   i ,  j    2
n
. For real constants a1 , , an , if  a Y is an
 , if i  j
i i
i 1

unbiased estimator of  1 , then

n n n n
(A)  a  0 and  a x  0
i 1
i
i 1
i i (B)  a  0 and  a x  1
i 1
i
i 1
i i

n n n n
(C)  ai  1 and  ai xi  0
i 1 i 1
(D)  ai  1 and  ai xi  1
i 1 i 1

Q. 11 – Q. 30 carry two marks each.
Q.11 Let ( X , Y ) have the joint probability density function

1 2  x
 y e , if 0  y  x  ,
f ( x, y )   2
0, otherwise.

Then P (Y  1| X  3) equals

1 1 1 1
(A) (B) (C) (D)
81 27 9 3

Q.12 Let X 1 , X 2 , be a sequence of i.i.d. random variables having the probability density function

 1
x 5 1  x  , 0  x  1,
3

f ( x)   B (6, 4)
0,
 otherwise.

Xi 1 n n U n  2 
Let Yi  and U n   Yi . If the distribution of converges to N  0,1 as
1 Xi n i 1 
n  , then a possible value of  is

(A) 7 (B) 5 (C) 3 (D) 1

MS 5/18

JAM 2016 MATHEMATICAL STATISTICS - MS

Q.13 Let X 1 , , X n be a random sample from a population with the probability density function

 4 e  4  x   , x   ,
f  x |    ,   .
0, otherwise

If Tn  min  X 1 , , X n  , then

(A) Tn is unbiased and consistent estimator of 
(B) Tn is biased and consistent estimator of 
(C) Tn is unbiased but NOT consistent estimator of 
(D) Tn is NEITHER unbiased NOR consistent estimator of 

Q.14 Let X 1 , , X n be i.i.d. random variables with the probability density function

e x , x  0,
f  x  
0, otherwise.

 
If X ( n )  max  X 1 , , X n  , then lim P X ( n )  log e n  2 equals
n

 0.5
(A) 1  e 2 (B) e  e
2
(C) e e (D) e e
2

Q.15 
Let X and Y be two independent N  0,1 random variables. Then P 0  X 2  Y 2  4 equals 
(A) 1  e  2 (B) 1  e  4 (C) 1  e  1 (D) e 2

Q.16 Let X be a random variable with the cumulative distribution function

 0, x  0,
x
 , 0  x  2,
8
F  x   2
x , 2  x  4,
 16
 1, x  4.


Then E  X  equals

12 13 31 31
(A) (B) (C) (D)
31 12 21 12

MS 6/18

JAM 2016 MATHEMATICAL STATISTICS - MS

Q.17 Let X 1 , , X n be a random sample from a population with the probability density function

1 x 
f  x  e , x  ,   0.
2

For a suitable constant K, the critical region of the most powerful test for testing H 0 :   1 against
H1 :   2 is of the form

n n
(A) X
i 1
i K (B) X
i 1
i K
n n
1 1
(C)  X
i 1
K (D)  X
i 1
K
i i

Q.18 Let X 1 , , X n , X n 1 , X n  2 , , X n  m  n  4, m  4  be a random sample from N   ,  2  ;
1 n 1 nm2
  ,   0. If X 1   i
n i 1
X and X 2   X i , then the distribution of the random
m  2 i  n 1
variable

X n  m  X n  m 1
T 
n nm2

 X  X     X  X 
2 2
i 1 i 2
i 1 i  n 1

is
(A) tn  m  2

2
(B) tn  m1
n  m 1

2
(C) tn  m  4
nm4

(D) tn  m  4

Q.19 Let X 1 , , X n  n  1 be a random sample from a Poisson   population,   0, and
n
T   X i . Then the uniformly minimum variance unbiased estimator of  2 is
i 1

T T  1 T T  1
(A) (B)
n2 n  n  1

T T  1 T2
(C) (D)
n  n  1 n2

MS 7/18

JAM 2016 MATHEMATICAL STATISTICS - MS

Q.20 Let X be a random variable whose probability mass functions f  x | H 0  (under the null
hypothesis H 0 ) and f  x | H1  (under the alternative hypothesis H1 ) are given by

X x 0 1 2 3
f  x | H0  0.4 0.3 0.2 0.1
f  x | H1  0.1 0.2 0.3 0.4

For testing the null hypothesis H 0 : X ~ f  x | H 0  against the alternative hypothesis
3
H1 : X ~ f  x | H1  , consider the test given by: Reject H 0 if X  .
2
If   size of the test and   power of the test, then

(A)   0.3 and   0.3
(B)   0.3 and   0.7
(C)   0.7 and   0.3
(D)   0.7 and   0.7

Q.21 Let X 1 , , X n be a random sample from a N 2  ,  2   population,   0. A consistent
estimator for  is

1 n 5 n 
1 2
(A)  Xi
n i 1 (B)   X i 2 
 n i 1 
1 n  1 n 2
1 2
(C) 
5n i 1
X i2 (D)   Xi 
 5n i 1 

Q.22 An institute purchases laptops from either vendor V1 or vendor V2 with equal probability. The
lifetimes (in years) of laptops from vendor V1 have a U  0, 4  distribution, and the lifetimes (in
years) of laptops from vendor V2 have an Exp 1 2  distribution. If a randomly selected laptop in
the institute has lifetime more than two years, then the probability that it was supplied by vendor V2
is
2 1 1 2
(A) (B) (C) (D)
2e 1 e 1  e 1 2  e 1

MS 8/18

JAM 2016 MATHEMATICAL STATISTICS - MS

Q.23 Let y ( x ) be the solution to the differential equation

dy
x4  4 x 3 y  sin x  0; y    1, x  0.
dx
 
Then y   is
2

10 1   4  12 1   4 
(A) (B)
4 4

14 1   4  16 1   4 
(C) (D)
4 4

Q.24 Let an  e  2 n sin n and bn  e  n n 2  sin n  for n  1. Then
2

 
(A)  a converges but  b does NOT converge
n 1
n
n 1
n

 
(B)  bn converges but  an does NOT converge
n 1 n 1
 
(C) both  a and  b converge
n 1
n
n 1
n

 
(D) NEITHER  an NOR  bn converges
n 1 n 1

Q.25 Let
 x sin 2 1 x  , x  0,  x  sin x  sin 1 x  , x  0,
f  x   and g  x  
0, x  0, 0, x  0.

Then
(A) f is differentiable at 0 but g is NOT differentiable at 0
(B) g is differentiable at 0 but f is NOT differentiable at 0
(C) f and g are both differentiable at 0
(D) NEITHER f NOR g is differentiable at 0

MS 9/18

JAM 2016 MATHEMATICAL STATISTICS - MS

Q.26 Let f :  0, 4   be a twice differentiable function. Further, let f  0   1, f  2   2 and
f  4   3. Then

1
(A) there does NOT exist any x1   0, 2  such that f   x1  
2
(B) there exist x2   0, 2  and x3   2, 4  such that f   x2   f   x3 
(C) f   x   0 for all x   0, 4 
(D) f   x   0 for all x   0, 4 

Q.27 Let f  x, y   x 2  400 x y 2 for all  x, y    2 . Then f attains its

(A) local minimum at  0, 0  but NOT at 1,1
(B) local minimum at 1,1 but NOT at  0, 0 
(C) local minimum both at  0, 0  and 1,1
(D) local minimum NEITHER at  0, 0  NOR at 1,1

Q.28 Let y  x  be the solution to the differential equation

d2y dy
4 2
 12  9 y  0, y (0)  1, y(0)   4.
dx dx

Then y 1 equals

1 3 2 3 3 2
(A)  e (B)  e
2 2

5 3 2 7 3 2
(C)  e (D)  e
2 2

Q.29 Let g :  0, 2   be defined by
x

g  x 
  x  t  e dt.
t

0
The area between the curve y  g   x  and the x-axis over the interval  0, 2 is

(A) e 2  1  
(B) 2 e 2  1


(C) 4 e 2  1  (D) 8  e  12

MS 10/18

JAM 2016 MATHEMATICAL STATISTICS - MS
  
Q.30 Let P be a 3  3 singular matrix such that P v  v for a nonzero vector v and

1  2 5 
P  0    0  .
 1   2 5
Then

(A) P 3 
1
5
7 P2  2 P 
(B) P 3   7 P 2  2 P 
1
4
(C) P   7 P 2  2 P 
3 1
3
(D) P 3   7 P 2  2 P 
1
2

MS 11/18

JAM 2016 MATHEMATICAL STATISTICS - MS

SECTION - B
MULTIPLE SELECT QUESTIONS (MSQ)
Q. 31 – Q. 40 carry two marks each.
Q.31 For two nonzero real numbers a and b, consider the system of linear equations

 a b   x  b 2 
b a   y   a 2 .
    

Which of the following statements is (are) TRUE?
(A) If a  b, the solutions of the system lie on the line x  y  1 2
(B) If a   b, the solutions of the system lie on the line y  x  1 2
(C) If a   b, the system has no solution
(D) If a   b, the system has a unique solution

Q.32 For n  1, let
n 2 n , if n is odd,
an    n
 3 , if n is even.

Which of the following statements is (are) TRUE?

(A) The sequence an  converges
(B) The sequence a n
1 n
 converges

(C) The series  a converges
n 1
n


(D) The series  a converges
n 1
n

Q.33 Let f :  0,     be defined by

 1 
f  x  x  e1 x 1  3 .
3

 x 

Which of the following statements is (are) TRUE?

(A) lim f  x  exists
x 

(B) lim x f  x  exists
x 

(C) lim x 2 f  x  exists
x 

(D) There exists m  0 such that lim x m f  x  does NOT exist.
x 

MS 12/18

JAM 2016 MATHEMATICAL STATISTICS - MS

Q.34 For x  , define f  x   cos  x    x 2  and g  x   sin  x  . Which of the following
statements is (are) TRUE?

(A) f  x  is continuous at x  2
(B) g  x  is continuous at x  2
(C) f  x   g  x  is continuous at x  2
(D) f  x  g  x  is continuous at x  2

Q.35 Let E and F be two events with 0  P  E   1, 0  P  F   1 and P  E | F   P  E  . Which
of the following statements is (are) TRUE?

(A) P  F | E   P  F 

 
(B) P E | F C  P  E 
(C) P  F | E   P  F 
C

(D) E and F are independent

Q.36 Let X 1 , , X n  n  1 be a random sample from a U  2  1, 2  1 population,   , and
Y1  min  X 1 , , X n  ,
Yn  max  X 1 , , X n  . Which of the following statistics is (are)
maximum likelihood estimator (s) of  ?
1
(A) Y1  Yn 
4
1
(B)  2 Y1  Yn  1
6
1
(C) Y1  3 Yn  2 
8
(D) Every statistic T  X 1 , , X n  satisfying
Yn  1  T X , , X  Y1  1
 1 n
2 2

Q.37 Let X 1 , , X n be a random sample from a N 0,  2   population,   0. Which of the
 n 
following testing problems has (have) the region  x1 , , xn    n :  x     as the most
2
i
2
n,
 i 1 
powerful critical region of level  ?

(A) H 0 : 2  1 against H1 : 2  2
(B) H 0 : 2  1 against H1 : 2  4
(C) H 0 : 2  2 against H1 : 2  1
(D) H 0 : 2  1 against H1 : 2  0.5

MS 13/18

JAM 2016 MATHEMATICAL STATISTICS - MS

Q.38 Let X 1 , , X n be a random sample from a N 0, 2  2   population,   0. Which of the
following statements is (are) TRUE?

(A)  X 1 , , X n  is sufficient and complete
(B)  X 1 , , X n  is sufficient but NOT complete
n
(C)  X is sufficient and complete
i 1
i
2

1 n 2
(D) 
2n i 1
X i is the uniformly minimum variance unbiased estimator for  2

Q.39 Let X 1 , , X n be a random sample from a population with the probability density function

 e  x , x  0,
f  x |    ,   0.
0, otherwise

Which of the following is (are) 100 1    % confidence interval(s) for  ?

   
 2  2 n,  2 
2   2 n,  
2

(A)   (B)  0, 
2 n ,1   2
,
 n n
  n

 2  Xi 2  Xi   2 Xi 
 i 1 i 1   i 1 
   n n

 2  2 n,  2 
2  2  X i 2  Xi 
(C)  n  (D)  2 
2 n ,1   2 i 1 i 1
, n , 2
    2 n ,  2  2 n ,1   2 
  Xi  Xi   
 i 1 i 1   

Q.40 The cumulative distribution function of a random variable X is given by

0, x  2,

1  7
F  x     x 2   , 2  x  3,
10  3
1, x  3.

Which of the following statements is (are) TRUE?

(A) F  x  is continuous everywhere
(B) F  x  increases only by jumps
1
(C) P  X  2  
6
 5 
(D) P  X  | 2  X  3  0
 2 

MS 14/18

JAM 2016 MATHEMATICAL STATISTICS - MS

SECTION – C
NUMERICAL ANSWER TYPE (NAT)
Q. 41 – Q. 50 carry one mark each.
Q.41 1 6
Let X 1 , , X 10 be a random sample from a N  3,12  population. Suppose Y1   X i and
6 i 1
Y  Y 
2
1 10
Y2  
4 i7
X i . If 1 2

has a 12 distribution, then the value of  is _______________

Q.42 Let X be a continuous random variable with the probability density function

2 x
 , 0  x  3,
f  x   9
0, otherwise.

 
Then the upper bound of P X  2  1 using Chebyshev’s inequality is ________________

Q.43 Let X and Y be continuous random variables with the joint probability density function

 x  y
e ,    x, y  0,
f  x, y   
0, otherwise.

Then P  X  Y   _____________________

Q.44 Let X and Y be continuous random variables with the joint probability density function

1   x2  y 2  2
f  x, y   e ,  x, y    2 .
2

Then P  X  0, Y  0   _________________________

Q.45  1
Let Y be a Bin  72,  random variable. Using normal approximation to binomial distribution,
 3
an approximate value of P  22  Y  28  is ________________________

MS 15/18

JAM 2016 MATHEMATICAL STATISTICS - MS

Q.46 Let X be a Bin  2, p  random variable and Y be a Bin  4, p  random variable, 0  p  1. If
5
P  X  1  , then P Y  1  ____________
9

Q.47 Consider the linear transformation

T  x, y , z    2 x  y  z , x  z , 3 x  2 y  z  .

The rank of T is ________________________

Q.48   1 
The value of lim n  e  n cos  4 n   sin    is _________________________
n 
  4 n 

Q.49 Let f :  0, 13   be defined by f  x   x13  e  x  5 x  6. The minimum value of the
function f on  0,13 is__________________________

Q.50 Consider a differentiable function f on  0,1 with the derivative f   x   2 2 x . The arc
length of the curve y  f  x  , 0  x  1, is ________________________

Q. 51 – Q. 60 carry two marks each.
Q.51 Let m be a real number such that m  1. If

m 1 1

   e dy dx dz  e 1,
y3

1 0 x

then m  ______________________

Q.52 Let
1  3 3 
P  0  5 6  .
0  3 4 
The product of the eigen values of P 1 is ________________________

MS 16/18

JAM 2016 MATHEMATICAL STATISTICS - MS

Q.53 The value of the real number m in the following equation

1 2  x2  2 2

   x  y  dy dx    r dr d
2 2 3

0 x m 0

is ________________________

Q.54 1
Let a1  1 and an  2  for n  2. Then
n
 
1 1 
  2  2 
n 1  an a n 1 
converges to ______________________

Q.55 Let X 1 , X 2 , be a sequence of i.i.d. random variables with the probability density
function

4 x 2 e  2 x , x  0,
f  x  
0, otherwise

n
 3n 
and let S n   X . Then lim P  S  2  3 n  is _____________________
i 1
i
n
n

Q.56 Let X and Y be continuous random variables with the joint probability density function

 c x2
 , 0  x  1, y  1,
f  x, y    y 3 ,
0,
 otherwise

where c is a suitable constant. Then E  X   ________________________

Q.57 Two points are chosen at random on a line segment of length 9 cm. The probability that the
distance between these two points is less than 3 cm is ______________________

MS 17/18

JAM 2016 MATHEMATICAL STATISTICS - MS

Q.58 Let X be a continuous random variable with the probability density function

 x 1
 , 1  x  1,
f  x   2
 0, otherwise.

1 1
Then P   X 2    ________________________
4 2

Q.59  1
If X is a U  0,1 random variable, then P  min  X , 1  X     _________________
 4

Q.60 In a colony all families have at least one child. The probability that a randomly chosen family from
this colony has exactly k children is  0.5  ; k  1, 2, . A child is either a male or a female with
k

equal probability. The probability that such a family consists of at least one male child and at least
one female child is _________

END OF THE QUESTION PAPER

MS 18/18