University of Hyderabad,
Entrance Examination, 2010
Ph.D. (Statistics-OR)

IHall Ticket No.

Answer Part A by circling the
Time: 2 hours Max. Marks: 75
correct letter in the array below: Part 25
Part 50

Instructions

1.
Calculators are not allowed.

2.
Part A carries 25 marks. Each cor­rect answer carries 1 mark and each wrong answer carries -0.33 mark. If you want to change any answer, cross out the old one and circle the

new one. Over written answers will be ignored.

3.
Part B carries 50 marks. Instructions for answering Part B are given at the beginning of Part B.

4.
Use a separate booklet for Part B.


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PART A

•
Find the correct answer and mark it on the answer sheet on the top page.

•
A right answer gets 1 mark and a wrong answer gets -0.33 mark.


1. Xl, ...,Xn are LLd random variables with absolutely continuous distribution
n
function then logF(Xi has
i=l

Normal distribution.


Beta distribution.


Gamma distribution.


Weibull distribution.



2. Let x be an observation from Bernoulli random variable taking values °and 1 with probabilities eand 1-Brespectively. If BE the ML estimate of is
x


x+2.
x+2

x+3



3. Let Xl, ...,Xn be a random sample from the Bernoulli distribution as described in Question an unbiased estimator for is
X-nX2

a
X-nX2

X+nX2

nX2-X


4. Let T be Binomial random variable with parameters nand E for each r ... n are equal to

n(n r ... n.


... ,n.



...,n.


... ,n.





u-5S­
5. Let Xl, X2 be a random sample from the population, define E[XIXl then is

t.



6. LetXll ..., Xn be a random sample from a distribution whose parameter is greater than and pdf

otherwise
Then MLE of is

Xitl+ 1].

1.






[n(log 1].
7. Let X be a non-negative continuous random variable with finite mean f.L. Let be the hazard rate of then

t.


f.L.


t·


t.



8. t 2 is Poisson process with parameter and let Yk be time till the kth arrival, P(Yk is equal to
1
6j=0 Jf.


e-l


e-l.


e-2 .


9. Consider the function xi -2X1X2 -2X2 2X1, X1,X2 E JR., let X2) denotes its Hessian, which of the following is correct?

X2) positive definite and hence X2) is convex.


X2) positive definite but nothing can be said about the convexity of the function.


X2) indefinite.


negative definite.


10. X Poisson(2) Y Poisson(3) and are independent. If X Y 10, the variance of X is
1;.


265'

11. X and Yare two random where P(X then which of the following is correct?










Nothing definite can be said.
12. For a random variable X with parameter if the functions and satisfy and 1 and for all x then is

1-0:2.


0:2·


cq+0<2
2
Q10<2
2

13. In a randomized block design of 4 treatments and 3 blocks the degrees of freedom of the residual sum of squares( Error sum of squares) is
3.

4.


5.


6.




14. A population of 30 units is divided into three strata with 12,12 units each. The number of different ways in which a stratified random sample of 5 units can be drawn in accordance to proportional allocation is

63 x 112•


5x 63xII.


20 x 122.


25x 11 x 12.

15. For any Gauss Markov model (YnxI X!3pXI, a with rank of X equal to p-l and n which of the following is correct?

Every component of !3pXI is certainly estimable.


Certainly no component of !3px 1 is estimable.


Every component of !3pXl may be non estimable.


Exactly one component of !3PXl is estimable.


16. In a hypothesis testing problem, the p-value was 0.06, which of the following is a correct decision?

The null hypothesis should be rejected at 0.05 level of significance.


The null hypothesis should be accepted at 0.05 level of significance.


The null hypothesis should be accepted at 0.07 level of significance.


None of the above.


17. Customers arrive in a super market in accordance with a homogeneous Poisson process, if the expected number of arrivals in one hour is 20, the expected length of time between the arrival times of the 6th and 7th customer is

3 minutes.


10 minutes.


5 minutes.


30 minutes.



18. X rv N20 where X2,... the diagonal elements of are
20 20
1 and the off diagonal elements are then YI L Xi and Y2 L aiXi are
i=l i=l
independently distributed if
20
L ai -20.
i=l

i=l
20

Lai
i=l

such a Y2 cannot be determined.
4


19. X is a random variable with probability distribution 1
P(X P(X

2
its characteristic function is

t


eit

.


sint.


cost.



20. Which of the following is always correct for any 5 x 5 real, skew-symmetric matrix

det O.


det A O.


A is singular.


A is definite.



21. Xi ... and are independent. In
is O.
.1
1S


is 1.



does not exist.


22. Xi, ... are independently distributed with the following distributions
P(Xi P(Xi P(Xi for i . P(Xi P(Xi for i .
n

Let Yn L Xi. Then
i=l


Yn 1 almost surely.


Yn 0 almost surely.


Yn 0 in probability but not almost surely.


Yn 1 in probability but not almost surely.



23. A fair die is rolled and then a fair coin is tossed as many times as the number that shows up on the die, the expected number of heads is

4.


3.





i.

24. The transition probability matrix of a Markov chain with state space S is
0 10)
001


100
Which of the following statements is not correct?


This Markov chain is irreducible.


This Markov chain has a stationary distribution.


This Markov chain is recurrent.


This Markov chain is aperiodic.


25. The multiple correlation coefficient between Y and Xl, X X3 is 0.98, and between Y and Xl, X2 is 0.91. The partial correlation coefficient between Y and X3 after removing the effects of Xl and X2 is in the interval

(0.3,0.4].


(0.5,0.7].


(0.7,0.8].


(0.8,0.9].



PARTB



•
There are 15 questions in this part. Answer as many as you can.

•
The maximum you can score is 50. Marks are indicated against each question.

•
The answers should be written in the separate answer script provided to you.


1.
An urn contains 10 balls of which X are red and the rest are white, where X takes values with probabilities each. 3 balls were drawn without replacement of which 2 were white and 1 was red, determine the probabilities that balls in the urn are red. marks]

2.
The conditional density function of X given Y y is


f(xl y)xY 0 x 1
Y 0 otherwIse

and the marginal distribution of Y is determine marks]
3. Candidates are allowed to appear for the civil service exams at most 4 times with the condition that he cannot write the exam if he has cleared in any of the earlier attempts. . For Ashok the probabilities of the clearing the exam in the·first attempt is 0.3, in the second it is 0.5, in the third it is 0.6 and in the fourth it is 0.8.
3rd

Determine the probability of failing in the pt, 2nd and the 4th at­tempts.


Expected number of attempts. marks]


4.
Let X and Y be independent standard normal random variables and let R and e be the polar coordinates of the vector Y). Show R2 and e are independent with R2 being exponential with mean 2 and e being uniformly distributed over marks]

5.
A random variable Y is said to have Weibull distribution with parameter 0 if its distribution function is given by




Define X what is the distribution of [4 marks]
6. Suppose that Xn is a random sample from a distribution with pdf x

o otherwise


where O. Find the UMVUE of using Rao-Blackwell-Lehmann-Scheffe theorem. marks]
7. {XnH" is a sequence of independent random variable with following distribu­tion
1
P(Xn P(Xn

n2 1
and

n2 P(Xn n2 1

Show that pU: as n 00, where Xl ... Xn and V(XI .,. X n marks]
8.
Let Xl, ..., Xn be a random sample from the e 0 population. Give an example of a Pivotal Quantity and use it to obtain a confidence interval of e. marks]

9.
Show that the Fisher information contained in a sample of size 4 from a Cauchy distribution with location parameter is 2. marks]

10.
(Xl,X2,X3,X4 and X rv where J is a 4x 4matrix in which every element is 1.

Compute E and D marks]
Show that IX2) rv [4 marks]

11.
A population consists of N units Ul, ... UN. X and Yare two variables of interest. Obtain an unbiased estimator for population covariance


N
I)Xi (Yi
i=l

based on a srswor of size n. Yi) are values of respectively for Ui ... N. marks]
12. The starting and current simplex tableaus of a given linear programming prob­lem( minimization problem) are given below. Find the values of unknowns ... marks]
Starting Tableaux Current Tableaux
Z Xl X2 X3 X4 X5 RHS
1 a 1 0 0 0
X4 0 b c d 1 0 6
X5 0 2 e 0 1 1



Z Xl X2 X3 X4 X5 RHS
1 0 j k I
Xl 0 g 2 2 1 0 f
X5 0 h i ""3 1 1 3


13. Consider the problem
minimize nL Xj
j=l
subject to nIIXj 1
j=l
Xj j ...


What are the KKT conditions for the problem?


Using the KKT conditions, find an optimal solution to the problem.


marks]
14. Let Xn be the size of a group that enters a restaurant at time n ... further are i.i.d random variables taking values ... with probabil­ities each. Now define a random variable Yn to be the size of the largest group that entered the restaurant till time n.

Show that the ...} is a Markov chain and write down its transition probability matrix.


Classify the states into communicating classes and identify the recurrent and transient classes.


For each transient state j and for each recurrent state k compute ijk.


[10 marks]
15. A sample of n units is selected from a population of N units without replace­ment in the following way. The first selection is made with unequal probabili­ties while the remaining n units are selected with equal probabilities. Let the first draw 8electioll probabilities be P1, .. ,PN for the population units U1, U2 .. UN respectively. Define
O. 1 if Ui is included in the sample 0 otherwise
Find I1i Probability that Ui is in the sample) and I1ij (Probability that Ui and Uj are in the sample) for each i,j ... N. marks]
9