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IIT JAM 2016 Mathematical Statistics (MS) Question Paper and Answer Key
JAM 2016: Mathematical Statistics Qn. No. Qn. Type Key(s) Mark(s) C 1 MCQ A 2 MCQ C 3 MCQ D 4 MCQ B 5 MCQ C 6 MCQ D 7 MCQ A 8 MCQ C 9 MCQ B 10 MCQ B 11 MCQ C 12 MCQ B 13 MCQ C 14 MCQ A 15 MCQ D 16 MCQ A 17 MCQ C 18 MCQ A 19 MCQ B 20 MCQ D 21 MCQ A 22 MCQ D 23 MCQ C 24 MCQ B 25 MCQ B 26 MCQ D 27 MCQ B 28 MCQ A 29 MCQ A 30 MCQ
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
JAM 2016: Mathematical Statistics Qn. No. Qn. Type Key(s) Mark(s) 31 MSQ A;B;D 32 MSQ A;C;D 33 MSQ A;B;C;D 34 MSQ B;D 35 MSQ A;C 36 MSQ A;B;C;D 37 MSQ A;B 38 MSQ B;C;D 39 MSQ A;B 40 MSQ C;D
2 2 2 2 2 2 2 2 2 2
JAM 2016: Mathematical Statistics Qn. No. Qn. Type Key(s) Mark(s) 41 NAT 4.9 to 5.1 42 NAT 0.49 to 0.51 43 NAT 0.49 to 0.51 44 NAT 0.24 to 0.26 45 NAT 0.52 to 0.62 46 NAT 0.80 to 0.81 47 NAT 1.9 to 2.1 48 NAT 0.24 to 0.26 49 NAT 4.9 to 5.1 50 NAT 2.1 to 2.2 51 NAT 3.9 to 4.1 52 NAT ‐0.51 to ‐0.49 53 NAT 0.24 to 0.26 54 NAT 0.74 to 0.76 55 NAT 0.97 to 0.98 56 NAT 0.74 to 0.76 57 NAT 0.5 to 0.6 58 NAT 0.20 to 0.21 59 NAT 0.49 to 0.51 60 NAT 0.3 to 0.4
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2
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www.myengg.com IIT JAM 2016 Mathematical Statistics (MS) Question Paper and Answer Key JAM 2016
MATHEMATICAL STATISTICS - MS
Special Instructions / Useful Data
Set of all real numbers
n
x ,, x : x , i 1,, n
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
1
n
i
The exponential distribution with probability density function
e x , x 0, f x | , 0 otherwise 0,
Exp
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
EX
Expectation of a random variable X
Var X
Variance of a random variable X
B m, n
1
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
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www.myengg.com IIT JAM 2016 Mathematical Statistics (MS) Question Paper and Answer Key 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) (B) (C) (D)
Q.2
4 3 2 1
Let , , be real numbers such that 0 and 0. Suppose
P
0
,
and P 1 P. Then (A) (B) (C) (D)
Q.3
0 0 0 0
(B) 15
(C) 16
(D) 17
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 (A)
MS
1 1 2 1
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
Q.4
and and and and
1 3
(B)
2 3
(C) 1
(D)
4 3
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www.myengg.com IIT JAM 2016 Mathematical Statistics (MS) Question Paper and Answer Key JAM 2016
Q.5
MATHEMATICAL STATISTICS - MS
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 (A) e 1 2
Q.6
(B)
3 1 2 e 2
(C)
1 1 2 e 2
e 1 2
Let E and F be two independent events with
P E | F P F | E 1, P E F Then P E equals (A)
Q.7
(D) e
1 3
(B)
1 2
(C)
2 and P F P E . 9
2 3
(D)
3 4
Let X be a continuous random variable with the probability density function
f ( x)
1 , x . (2 x 2 )3/2
Then E ( X 2 ) (A) equals 0 (C) equals 2
Q.8
(B) equals 1 (D) does not exist
The probability density function of a random variable X is given by
x 1 , f ( x) 0,
0 x 1 , 0. otherwise
Then the distribution of the random variable Y log e X 2 is (A) 22
Q.9
(B)
1 2 2 2
(C) 2 22
Let X 1 , X 2 , be a sequence of i.i.d. N (0,1) random variables.
(D) 12
Then, as n ,
1 n
n
X i 1
2 i
converges in probability to (A) 0 MS
(B) 0.5
(C) 1
(D) 2 4/18
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Q.10
MATHEMATICAL STATISTICS - MS
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 , if i j unbiased estimator of 1 , then n
(A)
a i 1 n
(C)
i
0 and
ai 1 and i 1
n
a x i 1
i
i
n
For real constants a1 , , an , if
n
0
(B)
a i 1
n
i
(D)
i 1
ai 1 and i 1
i
1
i
1
i
i 1
n
ai xi 0
n
a x
0 and
n
a x i
i 1
a Y i 1
i
i
is an
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 (A)
Q.12
1 81
(B)
1 27
(C)
1 9
(D)
1 3
Let X 1 , X 2 , be a sequence of i.i.d. random variables having the probability density function 3 1 x 5 1 x , 0 x 1, f ( x) B (6, 4) 0, otherwise.
Let Yi
Xi 1 and U n 1 Xi n
n
Y . i 1
i
If the distribution of
n , then a possible value of is (A)
MS
7
(B)
5
(C)
3
n U n 2
converges to N 0,1 as
(D) 1
5/18
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www.myengg.com IIT JAM 2016 Mathematical Statistics (MS) Question Paper and Answer Key JAM 2016
Q.13
MATHEMATICAL STATISTICS - MS
Let X 1 , , X n be a random sample from a population with the probability density function
4 e 4 x , x , , . f x | otherwise 0, 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 otherwise. 0,
If X ( n ) max X 1 , , X n , then lim P X ( n ) log e n 2 equals n
0.5 (B) e e 2 (D) e e
(A) 1 e 2 (C)
Q.15
e e
2
(A) 1 e 2
Q.16
Let X and Y be two independent N 0,1 random variables. Then P 0 X 2 Y 2 4 equals (B) 1 e 4
(C) 1 e 1
(D)
e 2
(D)
31 12
Let X be a random variable with the cumulative distribution function
0, x , 8 F x 2 x , 16 1,
x 0, 0 x 2, 2 x 4, x 4.
Then E X equals
(A)
MS
12 31
(B)
13 12
(C)
31 21
6/18
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www.myengg.com IIT JAM 2016 Mathematical Statistics (MS) Question Paper and Answer Key JAM 2016
Q.17
MATHEMATICAL STATISTICS - MS
Let X 1 , , X n be a random sample from a population with the probability density function
f x
1 e 2
x
, x , 0.
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
(A)
Xi K
(B)
n
1 K Xi
(D)
i 1
(C)
i 1
Q.18
n
Xi K
n
1 K Xi
i 1
i 1
Let X 1 , , X n , X n 1 , X n 2 , , X n m
, 0. If X 1
1 n
n
X i 1
n 4, m 4
and X 2
i
be a random sample from N , 2 ;
nm2
1 m2
i n 1
X i , then the distribution of the random
variable
T
X n m X n m 1 n
X i 1
X1 2
i
nm2
X
i n 1
i
X2
2
is (A) tn m 2
(B)
2 tn m1 n m 1
(C)
2 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
(A)
(C)
MS
T T 1 n2 T T 1 n n 1
(B)
(D)
T T 1 n n 1
T2 n2
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Q.20
MATHEMATICAL STATISTICS - MS
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 f x | H0
0
1
2
3
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
H1 : X ~ f x | H1 , consider the test given by: Reject H 0 if X If size of the test and power of the test, then (A) 0.3 (B) 0.3 (C) 0.7 (D) 0.7
Q.21
and and and and
3 . 2
0.3 0.7 0.3 0.7
5 (B) n
Xi i 1
Let X 1 , , X n be a random sample from a N 2 , 2
population, 0.
A consistent
estimator for is (A)
(C)
Q.22
1 n
1 5n
n
X i 1
i
n
X i 1
1 (D) 5n
2 i
n
1 2
2
X i 1 n
1 2
2 i
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 (A)
MS
2 2e
(B)
1 1 e
(C)
1 1 e 1
(D)
2 2 e 1
8/18
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www.myengg.com IIT JAM 2016 Mathematical Statistics (MS) Question Paper and Answer Key JAM 2016
Q.23
MATHEMATICAL STATISTICS - MS
Let y ( x ) be the solution to the differential equation
x4
dy 4 x 3 y sin x 0; y 1, x 0. dx
is 2
Then y
10 1 4
(A)
14 1 4
(C)
Q.24
(B)
4
(D)
4
4 16 1 4
4
Let an e 2 n sin n and bn e n n 2 sin n for n 1. Then 2
(A)
a
n 1
b
converges but
n
n 1
(B)
(C) both
an and
n 1
(D) NEITHER
a
Let
n 1
does NOT converge
n
does NOT converge
b
n 1
n
an NOR
n 1
n
bn converges but
n 1
Q.25
12 1 4
converge
b
n 1
n
converges
x sin 2 1 x , x 0, f x x 0, 0,
and
x sin x sin 1 x , x 0, g x x 0. 0,
Then (A) (B) (C) (D)
MS
f is differentiable at 0 but g is NOT differentiable at 0 g is differentiable at 0 but f is NOT differentiable at 0 f and g are both differentiable at 0 NEITHER f NOR g is differentiable at 0
9/18
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www.myengg.com IIT JAM 2016 Mathematical Statistics (MS) Question Paper and Answer Key JAM 2016
Q.26
MATHEMATICAL STATISTICS - MS
Let f : 0, 4 be a twice differentiable function. Further, let f 0 1, f 2 2
and
f 4 3. Then 1 2 (B) there exist x2 0, 2 and x3 2, 4 such that f x2 f x3
(A) there does NOT exist any x1 0, 2 such that f x1 (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
4
d2y dy 12 9 y 0, y (0) 1, y(0) 4. 2 dx dx
Then y 1 equals
Q.29
(A)
1 3 e 2
2
(B)
3 3 e 2
2
(C)
5 3 e 2
2
(D)
7 3 e 2
2
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
(C) 4 e 2 1
MS
8e
1
(B) 2 e 2 1 (D)
2
10/18
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Q.30
MATHEMATICAL STATISTICS - MS
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
1 7 P2 2 P 5 1 (B) P 3 7 P 2 2 P 4 1 3 (C) P 7 P 2 2 P 3 1 (D) P 3 7 P 2 2 P 2 (A) P 3
MS
11/18
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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 b a y a
2 . 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
1 n
n
converges
(C) The series
a
n 1
n
converges
(D) The series
n 1
Q.33
an converges
Let f : 0, be defined by
f x x e1
x3
1
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
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Q.34
MATHEMATICAL STATISTICS - MS
For x , define f x cos x x 2 statements is (are) TRUE?
and g x sin x .
Which of the following
(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
PF | E PF
(B) P E | F C P E (C)
C
(D) E and F are independent
Q.36
n 1 Y1 min X 1 , , X n ,
Let X 1 , , X n
be a random sample from a U 2 1, 2 1 population, , and
Yn max X 1 , , X n . maximum likelihood estimator (s) of ? 1 Y1 Yn 4 1 (B) 2 Y1 Yn 1 6 1 (C) Y1 3 Yn 2 8
Which of the following statistics is (are)
(A)
(D) Every statistic T X 1 , , X n satisfying
Q.37
Yn 1 T 2
Let X 1 , , X n be a random sample from a N 0, 2
X 1 , , X n
powerful critical region of level ?
2
population, 0.
following testing problems has (have) the region x1 , , xn n :
Y1 1
n
x i 1
2 i
Which of the
n2, as the most
(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
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www.myengg.com IIT JAM 2016 Mathematical Statistics (MS) Question Paper and Answer Key JAM 2016
Q.38
MATHEMATICAL STATISTICS - MS
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) (B)
X1 ,, X n is sufficient and complete X1 ,, X n is sufficient but NOT complete n
(C)
X i 1
(D)
Q.39
1 2n
2 i
is sufficient and complete
n
X i 1
2 i
is the uniformly minimum variance unbiased estimator for 2
Let X 1 , , X n be a random sample from a population with the probability density function
e x , x 0, f x | , 0. otherwise 0, Which of the following is (are) 100 1 % confidence interval(s) for ?
2 2 2 n, 2 2 n ,1 2 , (A) n n 2 Xi 2 Xi i 1 i 1 2 2 2 n, 2 2 n ,1 2 , n (C) n Xi Xi i 1 i 1 Q.40
2 2 n, (B) 0, n 2 Xi i 1 n n 2 X 2 Xi i i 1 i 1 , 2 (D) 2 2 n , 2 2 n ,1 2
The cumulative distribution function of a random variable X is given by
x 2, 0, 7 1 F x x 2 , 2 x 3, 3 10 1, x 3. Which of the following statements is (are) TRUE? (A) F x is continuous everywhere (B) F x increases only by jumps (C) P X 2
(D) P X
MS
1 6
5 | 2 X 3 0 2
14/18
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MATHEMATICAL STATISTICS - MS
SECTION – C NUMERICAL ANSWER TYPE (NAT)
Q. 41 – Q. 50 carry one mark each. Q.41
Let X 1 , , X 10 be a random sample from a N 3,12 population. Suppose Y1
Y2
Q.42
Y1 Y2
1 10 X i . If 4 i7
1 6 X i and 6 i 1
2
has a 12 distribution, then the value of is _______________
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 , x, y 0, e f x, y otherwise. 0,
Then P X Y _____________________
Q.44
Let X and Y be continuous random variables with the joint probability density function
f x, y
1 x2 y 2 2 , e 2
x, y 2 .
Then P X 0, Y 0 _________________________
Q.45
MS
1 random variable. Using normal approximation to binomial distribution, 3 an approximate value of P 22 Y 28 is ________________________ Let Y be a Bin 72,
15/18
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Q.46
MATHEMATICAL STATISTICS - MS
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 is _________________________ 4 n
The value of lim n e n cos 4 n sin n
Q.49
13 x Let f : 0, 13 be defined by f x x e 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 1 0
y3
dy dx dz e 1,
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
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www.myengg.com IIT JAM 2016 Mathematical Statistics (MS) Question Paper and Answer Key JAM 2016
Q.53
MATHEMATICAL STATISTICS - MS
The value of the real number m in the following equation 1
2 x2
x 0
2
y
2
x
2
2
m
0
dy dx r 3 dr d
is ________________________
Q.54
Let a1 1 and an 2
1 for n 2. Then n 1 1 2 2 a n 1 n 1 an
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 otherwise 0, and let S n
Q.56
n
X . i 1
i
Then lim P S n n
3n 2
3 n is _____________________
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
MS
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 ______________________
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Q.58
MATHEMATICAL STATISTICS - MS
Let X be a continuous random variable with the probability density function
x 1 , 1 x 1, f x 2 0, otherwise. 1 1 X 2 ________________________ 2 4
Then P
1 _________________ 4
Q.59
If X is a U 0,1 random variable, then P min X , 1 X
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
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