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# Indian Institute of Technology Guwahati (IIT-G) 2006 JAM Mathematical Statistics - Question Paper

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JAM 2006 Mathematical Statistics
Full ques. Paper in attachment

JAM 2006

MATHEMATICAL STATISTICS TEST PAPER

Special Instructions / Useful Data

1.    For an event A, P (A) denotes the probability of the event A.

2.    The complement of an event is denoted by putting a superscript c on the event, e.g. Ac denotes the complement of the event A.

3.    For a random variable X, E(X) denotes the expectation of X and V (X) denotes its variance.

4.    N(, a2) denotes a normal distribution with mean u and variance a2.

5.    Standard normal random variable is a random variable having a normal distribution with mean 0 and variance 1.

6.    P(Z > 1.96)= 0.025, P(Z > 1.65)= 0.050, P(Z > 0.675)= 0.250 and P(Z > 2.33)= 0.010 , where Z is a standard normal random variable.

7.    P(( > 9.21) = 0.01, P(( > 0.02) = 0.99, P(x3 > 11.34) = 0.01,    P(( > 9.49) = 0.05,

P(( > 0.71) = 0.95, P(52 > 11.07) = 0.05 and P(x52 > 1.15) = 0.95, where P(( >c) = a, where xi

has a Chi-square distribution with n degrees of freedom.

8.    n! denotes the factorial of n.

9.    The determinant of a square matrix A is denoted by | A|.

10.    R: The set of all real numbers.

11.    R: n-dimensional Euclidean space.

12.    y' and y" denote the first and second derivatives respectively of the function y(x) with respect to x.

NOTE: This Question-cum-Answer book contains THREE sections, the Compulsory Section A, and the Optional Sections B and C.

Attempt ALL questions in the compulsory section A. It has 15 objective type questions of six marks each and also nine subjective type questions of fifteen marks each.

Optional Sections B and C have five subjective type questions of fifteen marks each.

Candidates seeking admission to either of the two programmes, M.Sc. in Applied Statistics & Informatics at IIT Bombay and M.Sc. in Statistics & Informatics at IIT Kharagpur, are required to attempt ONLY Section B (Mathematics) from the Optional Sections.

Candidates seeking admission to the programme, M.Sc. in Statistics at IIT Kanpur, are required to attempt ONLY Section C (Statistics) from the Optional Sections.

You must therefore attempt either Optional Section B or Optional Section C depending upon the programme(s) you are seeking admission to, and accordingly tick one of the boxes given below.

 B C

Optional Section Attempted

The negative marks for the Objective type questions will be carried over to the total marks.

Write the answers to the objective questions in the Answer Table for Objective Questions provided on page MS 11/63 only.

1. If an > 0 for n > 1 and lim (an)n = L < 1, then which of the following series is not convergent?

n w '

(A)    y.4

n=1

n=1

w

2 n n=1

w

(B)     a

w __

(C) ZJa

n=1

i

(D) 2

n=1 */ at

2.    Let E and F be two mutually disjoint events. Further, let E and F be independent of G. If p = P(E) + P(F) and q = P(G), then P(EuF u G) is

(A)    1 - pq

(B)    q + p2

(C)    p+q2

(D)    p+q - pq

3.    Let X be a continuous random variable with the probability density function symmetric about 0. If

V (X) < w, then which of the following statements is true?

(A)    E (| X |) = E (X)

(B)    V (| X |) = V (X)

(C)    V(| X |) < V (X)

(D)    V (| X |) > V (X)

4.    Let

f (x) = X | X | + | x -1|, -w< x <w.

Which of the following statements is true?

(A)    f is not differentiable at x = 0 and x = 1.

(B)    f is differentiable at x = 0 but not differentiable at x = 1.

(C)    f is not differentiable at x = 0 but differentiable at x = 1.

(D)    f is differentiable at x = 0 and x = 1.

5.    Let Ax = b be a non-homogeneous system of linear equations. The augmented matrix [A : b] is given by

 " 1 1 -2 1 1" -1 2 3 -1 i 0 0 3 1 0 -1

Which of the following statements is true?

(A)    Rank of A is 3.

(B)    The system has no solution.

(C)    The system has unique solution.

(D)    The system has infinite number of solutions.

6. An archer makes 10 independent attempts at a target and his probability of hitting the target at each attempt

is 5. Then the conditional probability that his last two attempts are successful given that he has a total of 7

6

successful attempts is

1 55

(B)

15

(C)

36

x 7 s \ 3

5 Y (1Y

(D)

3! 5!

7.    Let

f (x) = (x- 1)(x-2)(x-3)(x-4)(x-5), -ro < x < ro.

The number of distinct real roots of the equation df (x) = 0 is exactly

dx

(A) 2    (B) 3    (C) 4    (D) 5

8.    Let

f(\ k |x| f (x )=/, , , -ro< x <ro (1 +1 x |)

Then the value of k for which f(x) is a probability density function is

(A) 6

(b) 2

(C)    3

(D)    6

f \    3f I 812

9.    If MX (f) = e    is the moment generating function of a random variable X, then P (- 4.84 < X < 9.60) is

(A) equal to 0.700

(b) equal to 0.925

(C) equal to 0.975 (d) greater than 0.999

10. Let X be a binomial random variable with parameters n and p, where n is a positive integer and

0 < p < 1. If a = P (x - np | >4n), then which of the following statements holds true for all n and

P ?

(A)    0 < a <

4

(B)     < a <

4    2

(C)    1 < a < 3

2    4

3

(D)    - < a < 1 v 7 4

11. Let X, X2,...,Xn be a random sample from a Bernoulli distribution with parameter p; 0 p < 1. The

47,+2]Tx,

bias of the estimator -i=Ls for estimating p is equal to

(A)    J+1 Vp - 2,

(B)    ~Lr f 2 - p

n + vn v 2 ,

1

(C)

4n +1 V 2 yfn

1 ( 1

(D)

-p

4n +1 V2

12. Let the joint probability density function of X and Y be

f (x,,)={e"x if 0 < ' <x

10, otherwise.

 Then E (X) is (A) 0.5 (B) 1 (C) 2 (D) 6

t

1, t = 0.

1 *3

Then the value of lim f f (t) dt

x0 v2 J V

X X 2

X2

(A)    is equal to -1

(B)    is equal to 0

(C)    is equal to 1

(D)    does not exist

14. Let X and Y have the joint probability mass function;

1

P (( = x, Y = y ) =

x, y = 0,1,2,....

2y+2(y +1)I 2y + 2

Then the marginal distribution of Y is

(A)    Poisson with parameter A = 4

(B)    Poisson with parameter A = 2

(C)    Geometric with parameter p = -4

(D)    Geometric with parameter p = 2

_ 1 3

15. Let Xj, X2 and X3 be a random sample from a N(3, 12) distribution. If X = Xi and

3 =1

1 3 2

S2 = '(Xi -X) denote the sample mean and the sample variance respectively, then

=1

P (1.65 < X < 4.35, 0.12 < S2 < 55.26) i

(A) 0.49

(b)    0.50

(c)    0.98

(D) none of the above

is

16. (a) Let X1, X2,Xn be a random sample from an exponential distribution with the probability density function;

\d e x, if x > 0,

f (x ;Q) =

v0,    otherwise,

where d> 0. Obtain the maximum likelihood estimator of P(X > 10) . 9 Marks

(b) Let X1, X2,..., Xn be a random sample from a discrete distribution with the probability mass function given by

P(X = 0)=P(X = 1) = 2; P( = 2) = f. 0<e< 1.

Find the method of moments estimator for Q.

6 Marks

17.    (a) Let A be a non-singular matrix of order n (n > 1),    with | A | = k . If adj(A) denotes the adjoint of the

matrix A, find the value of | adj(A) |.    6 Marks

(b) Determine the values of a, b and c so that (1, 0,    -1) and (0, 1, -1) are eigenvectors of the matrix,

' 2 1 1"

a 3 2.    9 Marks

3 b c

18.    (a) Using Lagranges mean value theorem, prove that

b - a    , 7 , b - a

< tan b - tan a <

1 + b

TT

where 0 < tan-1 a < tan-1 b < .

2

(b) Find the area of the region in the first quadrant that is bounded by y = Vx, y = x - 2 and the x - axis .

9 Marks

6 Marks

19. Let X and Y have the joint probability density function;

(x2 + 2 y2)

, if x > 0, y > 0, otherwise.

c x y e 0,

f (x y ) =

Evaluate the constant c and P(X2Y2).

20.    Let PQ be a line segment of length J3 and midpoint R. A point S is chosen at random on PQ. Let X, the distance from S to P, be a random variable having the uniform distribution on the interval (0, J3). Find the probability that PS, QS and PR form the sides of a triangle.

21.    Let X1, X2,..., Xn be a random sample from a N (, 1) distribution. For testing H0: /u = 10 against

1 n

H1 : u = 11, the most powerful critical region is Xk, where X = V Xi. Find k in terms of n such

n i=1

that the size of this test is 0.05. Further determine the minimum sample size n so that the power of this test is at least 0.95.

22.    Consider the sequence {sn}, n > 1, of positive real numbers satisfying the recurrence relation

sn-1 + sn = 2 sn+1 fr al1 n2 .

(a)    Show that | sn+1 - sn | = I s2 - s1 | for all n > 1.

(b)    Prove that {sn} is a convergent sequence.

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

0,    if x < 0,

1    (l + x3 ),    if 0 < x < 1,

F ( x) =

1    r    21

5    3 + (x -1) , if 1 < x < 2,

1,    if x > 2.

Find P(0 < X < 2), P(0 < X < 1) and P2 < X <

24.    Let A and B be two events with P (A | B) = 0.3 and P (A | Bc) = 0.4 . Find P(B | A) and P(Bc | A) in

terms of P(B). If P(B | A) < 1 and 1P(Bc | Ac) < , then determine the value of P(B).

4    3 4    16

Optional Section B

25.    Solve the initial value problem

y - y + y2 (x2 + 2 X + l) = 0, y(0) = 1.

26.    Let y( x) and y2( x) be the linearly independent solutions of

xy" + 2 yr + x ex y = 0.

If W(x) = y(x) y2(x) - y2(x) y(x) with W(1) = 2, find W(5).

1 1

27.    (a) Evaluate J J x2 exy dxdy.    9 Marks

0 y

(b) Evaluate JJJ z dxdy dz, where W is the region bounded by the planes x = 0, y = 0, z = 0, z = 1

W

and the cylinder x2 + y2 = 1 with x > 0, y > 0.

6 Marks

28.    A linear transformation T: 3 2 is given by T (x, y, z) = (3 x + 11 y + 5 z, x + 8 y + 3 z).

Determine the matrix representation of this transformation relative to the ordered bases {(1, 0,1), (0,1,1), (1, 0, 0), {(1,1), (1, 0)}. Also find the dimension of the null space of this transformation.

0,    if x + y = 0.

Determine if f is continuous at the point (0, 0) .    6 Marks

(b) Find the minimum distance from the point (1, 2, 0) to the cone z2 = x2 + y2.    9 Marks

Optional Section C

30. Let X1, X2,..., Xn be a random sample from an exponential distribution with the probability density function;

1 --

e 0, if x > 0,

f (x; 0) =

e

0,    otherwise,

where e > 0. Derive the Cramer-Rao lower bound for the variance of any unbiased estimator of e.

1 n

Hence, prove that T = V Xi is the uniformly minimum variance unbiased estimator of e.

n i=1

31. Let Xj, X2,... be a sequence of independently and identically distributed random variables with the probability density function;

f (x) =

1    x2 e- x,

2

0,

if x > 0, otherwise.

Show that lim P ( + ... + Xn > 3 (" - )) > )

32. Let Xj, X2,..., Xn be a random sample from a N (u, a2) distribution, where both u and a2 are

unknown. Find the value of b that minimizes the mean squared error of the estimator

2

b n    _    _ 1 n

T =-V ( -X) for estimating a2, where X = Xi.

1 i=1    n ,=1

33.    Let Xj, X2,..., X5 be a random sample from a N(2, a2) distribution, where a2 is unknown. Derive the most powerful test of size a = 0.05 for testing H0:a2 = 4 against Hl:a2 = 1.

34.    Let Xj, X2,..., Xn be a random sample from a continuous distribution with the probability density function;

2

x

2 x

T

, if x > 0,

f (x; ) =

0,    otherwise,

where A > 0. Find the maximum likelihood estimator of A and show that it is sufficient and an unbiased estimator of A.

8 