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 Students 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

EX 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
Let X 1 , X 2 , be a sequence of i.i.d. N (0,1) random variables. Then, as n ,
n
X
i 1
i
2

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
i 1
i 0 and a x
i 1
i i 0 (B) a
i 1
i 0 and a x
i 1
i i 1
n n n n
(C) ai 1 and
i 1
ai xi 0
i 1
(D) ai 1 and
i 1
a x
i 1
i 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 Y . If the distribution of converges to N 0,1 as
1 Xi n i 1
i

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
e e (D) e e
2
(C)

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
f x

, x , 0.
x
e
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)
i 1
Xi K (B)
i 1
Xi K
n n
1 1
(C)
i 1 Xi
K (D)
i 1 Xi
K

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 ;

n nm2
1 1
, 0. If X 1
n
X
i 1
i and X 2
m2

i n 1
X i , then the distribution of the random

variable

X n m X n m 1
T
n nm2

X X1 X X2
2 2
i i
i 1 i n 1

is
(A) tn m 2

2
(B) tn m1
n m 1

2
(C) tn m 4
nm4

(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
n 1 2
1 5
X
n
(A)
n i 1
i (B) Xi 2

n i 1
n 1 2
1 1
X
n

2
(C) i (D) X 2
5n i 1
i
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)
2e 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
n 1
n converges but b
n 1
n does NOT converge

(B) bn converges but
n 1
a
n 1
n does NOT converge

(C) both an and
n 1
b
n 1
n converge

(D) NEITHER an NOR
n 1
b
n 1
n converges

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 3 3
(A) e 2
(B) e 2
2 2

5 3 7 3
(C) e 2
(D) e 2
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) 8e 2
1

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
n 1
n converges

(D) The series
n 1
an converges

Q.33 Let f : 0, be defined by

1
f x x e1 x3
1 .
x3

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) PF | E PF
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 Y1 1
X 1 , , X 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 2
i n2, as the most
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) X1 ,, X n is sufficient and complete

(B) X1 ,, X n is sufficient but NOT complete
n
(C) X
i 1
i
2
is sufficient and complete

n
1
(D)
2n
X
i 1
i
2
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
Y1 Y2
2
1 10
Y2 X i . If
4 i7
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 Chebyshevs 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
y3
dy dx dz e 1,
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 3 dr d
2 2

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 .
i 1
i Then lim P S n
n
2
3 n is _____________________

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