University of Hyderabad,
Entrance Examination, 2012

Ph.D. (Statistics-OR)
I Hall Ticket No. 1
Answer Part A by circling the correct letter in the array below:
Time-2 Hours Max. Marks: 75 Part A:25 Part B:50
Instructions
1 a b c d
2 a b c d
3 a b c d
4 a b c d
5 a b c d

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.


6 a b c d
7 a b c d
8 a b c d
9 a b c d
10 a b c d

11 a b c d
12 a b c d
13 a b c d
14 a b c d
15 a b c d

16 a b c d
17 a b c d
18 a b c d
19 a b c d
20 a b c d

21 a b c d
22 a b c d
23 a b c d
24 a b c d
25 a b c d




Part-A

• Find the correct answer and mark it on the OMR sheet. Each correct answer gets 1 mark and wrong answer gets -0.33 marks..
1. Two squares are chosen at random on a chess board. What is the probability that
they have one common side?
1 32 491

­
2016 64 36
18
. 2. Let Xl, X 2X X4 be independent random variables. X 2,X X4 have Poisson dis­
4
tribution with mean further Y Xi ,...., Poisson(25). The distribution of Xl i=l
is
Binomial Exponential Rectangular Poisson
3.
Let Xl,X2 be i.i.d. random variables with pdf define YI min{Xl X and Y2 the joint pdf of Yi and Y2 is

4.
For two events A and P(AIB) so

5.
Let Xl and X2 be two independent observations of a Bernoulli random variable that takes values 1 or 0 with probabilities eand respectively. If eE the maximum likelihood estimate of eis




XI ;X2 XI: 2X2
6. Xl, X2, X3 is a random sample from the define YI Xl X2 X then



7. Five numbers are drawn from the set ... 100} by SRSWOR, the probability p that their median is at least 20 is
less than 0.25 p 0.5 p 0.8 p 0.99
8. X rv and Y then is
5 6 10 12
9. If for a random variable 1 and E(X2 then
0.5 0.75
X 0.6 X 0.5
10. lim where an .!ris equal to
_00n n

11. Based on a random sample of size 16 from the N(p" population. The 95% confidence interval for was It means

the mean of this random variable is certainly in the interval


one is 95% sure that the mean is in the interval


the mean is greater than 52 with probability 0.025


none of the above


12. In a simple regression of Y on what is the correlation coefficient between Y and Y-the predicted value of Y if the correlation between X and Y is
0
13. Xl, ... Xn is a random sample from the U(a b a O. Let min(XI, ... ,Xn max(XI, ... and X txi the joint
n i=l
sufficient statistic for is


14. Xl and X 2 are i.i.d. random variables, then Xl X 2 andXl 2

have different expected values are uncorrelated but not independent


are independent have different variances


15.
If the characteristic function of i.i.d. random variables Xl and X 2 is where ¢ R then the characteristic function of Xl 2 is

16.
Xl, ... n is a random sample from theN(p, population. IfTI Xl +...



n
and T2 Xf +... then which of the following statements is correct regarding sufficient statistics for p and (72

TI is sufficient statistic for p


T2 is sufficient statistic for (72


T2 -TI is sufficient for


(Til T2) is sufficient for


17. A is an interval and B is an interval then A6B is

a closed interval


an open interval


an open set that is not an interval


a finite set



18. The one step transition probability matrix of a homogeneous Markov chain
{Xm n is
1/3 1/6 1/6
1/4 1/4 1/4 1/4
1/5 1/4 1/4 3/10 .


1/6 1/3 1/3

Then which of the following is not correct?
P(X7 llXs P(Xll llXg this Markov chain is
irreducible the states 1 and 2 are the only recurrent states it is a recurrent Markov chain
19. Each of the seven treatments have to appear in blocks of sizes four each. Which of the following choices on "number of blocks" and "number of blocks in which a pair of treatment appear" respectively, gives a valid BIBD.
6,3 7,2 8,3 7,1
20. In a factorial design, it is decided to confound the effect ABCD. Blocks 1 and 2 in a replication contain the following combinations or treatments Block abc abc d abd Block ab ac bc bd cd abcd What are the other treatments in blocks 1 and 2 respectively?
and ad} bed} and ad} and {acd,bcd} {acd,bcd} and
21. is a sequence of independent random variables with pmf P(Xn 12
P(Xn n2 P(Xn 1-n2 1· Let Sn Xl ... Xn. Then

P (ISn/nl E for infinitely many n 0


P (ISn/nl for infinitely many n 1



E 0


lim P (ISn/nl 1/2




10 5
22. The dispersion matrix of a random vector X X is 59 a . The value

5 a 16
of sothatXl X2+ X3andXl-2X2 X3 are uncorrelated, is

8 21 13 16
23. In a complete randomized design, for three treatments whose effects are which of the following is testable?




0




2­





24. Shoppers arrive at a mall in accordance with a homogeneous Poisson Process, if the expected number of arrivals in an hour is 600, the expected time between consecutive arrivals is

1 min


10 second


6 second


different between different consecutive pairs of arrivals


25. Xn r-..J n ... then


Xn 0 in probability but Xn 0 weakly


Xn 0 weakly


Xn 0 in probability




Part-B
• Answer as much as you can,the maximum marks that you can score is 50.
1.
7 girls and 8 boys are randomly arranged in a row. Determine the probability

of the event that no two girls are sitting together.
b)Two numbers are drawn from the set ... 100} by SRSWOR, determine the
probability that the largest of the two is a prime number.


2.
Xl and X2 are i.i.d. exp(A) random variables. Determine E(XIIXI +X2 10) and





3. 2.8,3.2,4.1,2.2,1.8,2.7 are six independent observations of a random variable X with pdf
1
f.L, :xe f.L
where A 0 and -00 f.L 00. Assume f.L 1 and construct a 90% confidence interval for A based on the given sample. 5
4. LetXll ... ,Xn be i.i.d. with pdf
exfJ 0 x e 0
Find the MLE of e.
Find the asymptotic distribution of the MLE.

5.
In a bag there are N slips numbered ... N not known. Draw a SRSWR of size n. Let Xl,"" Xn denote the numbers drawn, obtain the most powerful level a test for Ho N No vs. N No, N No

6.
A school is preparing a trip for 400 students. the company who is providing the transportation has 10 buses of 50 seats each and 8 buses of 40 seats, but only 9 drivers available. The rental cost for a large bus is Rs. 8000 and Rs. 6000 for small bus. Calculate how many buses of each type should be used for the trip for at least possible cost.

7.
Find the optimal solution of the dual of the following LPP{Pl) and hence the solution to this PI.


5 10

PI: maximize Xl +X2 +2X3
Subject to Xl +2X2 2XI +X2 +2X3 1 and all Xi 0 10
8. are independent random variables with the following probability distribu­tion:
P{Xj P{Xj P{Xj j ...

Can you say that Xn 0 in probability?

ii) Xn 0 with probability
Explain or prove as the case may be.
What can you say about the limiting distribution (as n of Tn 1 n

-LXj Rl

nV2 j=l