How To Exam?

a knowledge trading engine...


Indian Institute of Technology Guwahati (IIT-G) 2005 JAM Maths Statistics - Question Paper

Wednesday, 23 January 2013 06:50Web

Page 1 of 2

JAM 2005 Maths Statistics
Full ques. Paper in attachment

A

2005 - ST

Test Paper Code : ST Time : 3 Hours    Max. Marks : 300

INSTRUCTIONS

1.    The qucstion-cum-answer book has 40 pages and has 29 questions. Please ensure that the copy of the question-cum-answer book you have received contains all the questions.

2.    Write your Roll Number, Name and the name of the Test Centre in the appropriate space provided on the right side.

3.    Write . the answers to the objective questions against each Question No. in the Answer Table for Objective Questions, provided on page No. 9. Do not write anything else on this page.

4.    Each objective question has 4 choices for its answer: (A), (B), (C) and (D). Only ONE of them is the correct answer. There will be negative marking for wrong answers to objective questions. The following marking scheme for objective questions shall be used :

(a)    For each objective question, you will be awarded 6 (six) marks if you have written only the correct answer.

(b)    In case you have not written any answer for a question 3?ou will be awarded

0 (zero) mark for that question.

(c)    In all other cases, you will be awarded -2 (minus two) marks for the question.

5.    Answer the subjective question only in the space provided after each question.

6.    Do not write more than one answer for the same question. In case you attempt a subjective question more than once, please cancel the answer(s) you consider wrong. Otherwise, the answer appearing later only will be evaluated.

7.    All answers must be written in blue/ black/blue-black ink only. Sketch pen, pencil or ink of any other colour should not be used.

8.    All rough work should be done in the space provided and scored out finally.

9.    No supplementary sheets will be provided to the candidates.

10.Logarithmic    Tables / Calculator of any kind / cellular phone / pager / electronic gadgets are not allowed.

11.The    question-cum-answer book must be returned in its entirety to the Invigilator before leaving the examination hall. Do not remove any page from this book.

12.Refer    to special instruction/useful data on the reverse.

A

2005 - ST

READ THE INSTRUCTIONS ON THE LEFT SIDE OF THIS PAGE CAREFULLY

ROLL NUMBER

Name :

Test Centre :

Do not write your Roll Number or Name anywhere else in this question-cum-answer book.

I have read all the instructions and shall abide by them.

Signature of the Candidate

I have verified the information filled by the Candidate above.

Signature of the Invigilator

NOTE : Attempt ALL the 29 questions. Questions 1-15 (objective questions) carry six marks each and questions 16 - 29 (subjective questions) carry fifteen marks each.

Write the answers to the objective questions ONLY in the Answer Table for Objective Questions provided on page 9.

1. Let

1

2

3

4

5

1

2

3

4

8

6

3

2

4

6

7

10

3

4

7

10

14

16

7

Then the rank of the matrix P is

(A)    1

(B)    2

(C)    3

(D)    4

Six identical fair dice are thrown independently. Let S denote the number of dice showing even numbers on their upper faces. Then the variance of the random variable S is

<*> !

(B)    1

(C)    |

(D)    3

4. Let Xlt X2, X21 be a random sample from a distribution having the variance 5. Let _ 1 21 21 _

X = YjXi and S =    - X)1. Then the value of E(S) is

21 ' . 1 =i    j=i

(A)    5

(B)    100

(C)    0.25

(D)    105

(A)    chi-square with 2 degrees of freedom

(B)    chi-square with 1 degree of freedom

(C)    F with (2,2) degrees of freedom

(D)    F with (1,1) degrees of freedom

In three independent throws of a fair dice, let X denote the number of upper faces showing six. Then the value of E(3 - X)2 is

1

0

1 + X

1 + X

0

1

1

1

1

1 + X

0

1 + X

1

1 + X

1+ X

0

Then the determinant of the matrix P is

A.


(A)    3(x +1)3

(B)    3(x+l)2

(C)    3(x +1)

(D)    (x + l)(2x + 3)

f    3    3]

8. The area of the region Ux,y): 0 < x, j/ < 1, <x + y < > is

I    4    2 J

9. Let E, F and G be three events such that the events E and F are mutually exclusive,

P(EUF) = 1, P(E f]G) = and P(G) = . Then P(F n G) equals

4    12

(A)

(B)

(C)

(D)


12

1

4

_5_

12

1

3


10. Let X and Y have the joint probability mass function

P(X =x,Y =    y=l,2,...,x; x =1,2,3.

Then the value of the conditional expectation E (Y\X = 3) is

(A)    1

(B)    2

(C)    1.5

(D)    2.5

11. Let X1 and X2 be independent random variables with respective moment generating functions

AfjCi) = + -j-e* j and M2(t) = e2(-e _1), - o < t <

Then the value of P(Xx + X2 =1) is

(A)    e~2 64

(B)    e~2 64

(C)     e~2 64

(D)     e~2 32

.A l-i 2 t2 dt


I e

n \ PT n+v2n


lim


12.


equals


22 r


(A)    0.5

(B)    0

(C)    0.0228

(D)    0.1587

13. Let X1 and X2 be two independent random variables having the same mean 0. Suppose that E(X1 - 0)2 =1 and E(X2-0)2=2. For estimating 0, consider the estimators Ta(X1,X2) = aX-l +(l-a)X2, ore [0,1]. The value of a e [0,1], for which the variance of Ta(Xl,X2) is minimum, equals

(A)

(B)

(C)

(D)

14. Let Xj =3, x2 -4, x3 =3.5, x4 =2.5 be the observed values of a random sample from the probability density function

f{x\e) = -


x > 0, 0 > 0


+ e


-e e +e9

n    2


Then the method of moments estimate (MME) of 0 is

(A)    3.5

(B)    4

(C)    2.5

(D)    1

Let xx = -2, x2 = 1, x3 = 3, x4 = -4 be the observed values of a random sample from the distribution having probability density function

f\s) = ~rj< -o<x<e, d>o.

e - e

Then the maximum likelihood estimate of 6 is

(A)    3

(B)    0.5

(C)    4

(D)    Any value between 1 and 2

Let X and Y be independent and identically distributed uniform random variables over the interval (0,1) and let S = X + Y. Find the probability that the quadratic equation

Find the number of real roots of the polynomial f(x) = x5 +x3 -2x +1.

2

1

1

1

n + 1 1

72+1

2

72+1

1

72+1

1

n + 1 1

n + 1 1

72+1

2

72+1

1

n +1

71+1

72+1

72+1

1

1

1

2

.71+1

n +1

72+1

72+1

Let I be the n x n identity matrix and let J be the nxn matrix whose all entries are 1. Express the matrix P as al +bJ, for suitable constants a and b. It is known that the inverse of P is of the form cl + dJ, for some constants c and d. Find the constants c and d in terms of n.

Let

f{x) = ix + 1) | X2 - 1 | , - oo < X < oo.

Verify the differentiability of the function /() at the points x = -1 and x = 1.

There are four urns labeled U1,U2, U3 and U4, each containing 3 blue and 5 red balls. The

fifth urn, labeled U5, contains 4 blue and 4 red balls. An urn is selected at random from

these five urns and a ball is drawn at random from it. Given that the selected ball is red, find the probability that it came from the urn U5.

Let

if    x < 0

if    0 < x < 1

if    1 < x < 2

if    2 < x < 3

if    x > 3.


0,

l 10

x + 2


F(x)


c (6x - x2 -1) 2

1,


Find the value of c for which F(-) is a cumulative distribution function of a random variable X. Also evaluate P( 1 < X < 2).

A.

Let A, B and C be pair-wise independent events such that P(A fl 5) = 0 1 and P(BnC) = 0-2.

Show that P(AC U C) > .

8

Let the random variables X and Y have the joint probability density function

ce~{x+y), y > x > 0

f(x,y) = *

0,    otherwise

Evaluate c and E(Y I X = 2).

24. Find the area of the region enclosed between the curves y = (x - 2)2 and y = 4 - x2.

Let X be the sample mean of a random sample of size 10 from a distribution having the probability density function

-X

fix | 6) - e 0 , x > 0, 6 > 0.

6

Using Chebyshevs inequality, show that PiO <X < 26) > 0 9.

Let X be a continuous random variable having the probability density function

-2 < x < 3


fix)


otherwise


(x + 2), 25v '

0,


Find the cumulative distribution function of Y =X2. Hence derive the expression for probability density function of Y.

Let X be a single observation from the probability density function fix | 6) = id + l)x, 0 < x < 1,

13

where 6 e {1,2} is unknown. Find the most powerful test of level a = for testing H0 : 6 = 1 against H1:0 = 2. What is the power of the test?

Let Xl7 X2, , Xn be a random sample from a population having the probability density function

-ix-O)

x > 0,

a

f(x\0,a) =


oo < o < oo a > 0.

otherwise,

Find the method of moments and maximum likelihood estimators of 0 and a.

Let

2

f(x) = 3(x-2)3 ~{x-2), 0<x<20.

Let x0 and j>0 be the points of the global minima and the global maxima, respectively, of /*() in the interval [0,20]. Evaluate f(x0) + f(y0).

ft*

Jt







V








1

Consider the following system of linear equations:

x + y + z = 3, x + az = b, y + 2z = 3.

This system has infinite number of solutions if

(A)    a = -1, 6 = 0

(B)    a = 1, b = 2

(C)    a = 0, b = 1

(D)    a = -1, 6 = 1


  • «
  •  Start 
  •  Prev 
  •  1 
  •  2 
  •  Next 
  •  End 
  • »





Attachment:

( 0 Votes )

Add comment


Security code
Refresh

Earning:   Approval pending.
You are here: PAPER Indian Institute of Technology Guwahati (IIT-G) 2005 JAM Maths Statistics - Question Paper