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DETACHABLE
STATISTICS
Paper I

ITime Allowed: Three Hours IMaximum Marks: 200 I
QUESTION PAPER SPECIFIC INSTRUCTIONS Please read each of the following instructions carefully before attempting questions. There are EIGHT questions in all out of which, FIVE are to be attempted. Question no. 1 5 are compulsory. Out of the remaining SIX questions, THREE are to be attempted selecting at least ONE question from each of the two Sections A and B. Attempts ofquestions shaH be counted in chronological order. Unless struck off, attempt ofa question shall becountedeven ifattemptedpartly.Anypage orportionofthepageleftblankintheanswerbook must be clearly struck off. All questions carry equal marks. The number ofmarks carried by a question/part is indicated against it. Answers must be written in ENGLISH only. Unless otherwise mentioned, symbols and notations have their usual standard meanings. Assume suitable data, ifnecessary and indicate the same clearly.
SECTION
1. Answer the following:
l.(a)A particular test for diabetes, when administered to people who have diabetes, can detect the disease correctly in
95% of the cases. When the test is administered to people who do NOT have diabetes, the test shows 'false positive'
in of the cases.
Assuming that of the population has diabetes, answer the following:
1.(a)(i)What is the probability that a randomly selected person will be declared to be diabetic by the test?
a person tests positive for diabetes, what is the probability that he is actually diabetic? 8
1.(b)Two fair dice are thrown. If X is the sum of the numbers shown up, use Chebychev's inequality to get an
upper bound for Also obtain the exact value of this probability. 8
l.(c)The distribution function of a bivariate random vector is
<img src='./qimages/1277-1c.jpg'>

1.(c)(i)Compute the marginal densities of X and Y.
l.(c)(ii)For what values of alpha are X and Y independent? 8
A random sample of size n is drawn from a distribution having p.d.f.
<img src='./qimages/1277-1d.jpg'>
l.(d)(i)What is the maximum likelihood estimator for 1/theta?
this estimator consistent? 8
Let Xl,X2, ..., Xn be i.i.d. with each Xi being a Bernoulli (theta) random variable. Obtain the
likelihood ratio test for testing H0:theta<=theta0 against H1:theta>theta0. 8

Let be a random vector.
2.(a)(i)Show that whenever X and Y are independent
2.(a)(ii)Give an example to show that the converse of above in false 10
2.(b)Give an example to show that convergence in probability does not imply convergence almost
surely.
2.(c)A random variable has characteristic function pie(t) What is its density function? 10
2.(d)The transition probability matrix of a Markov chain with state space is
<img src='./qimages/1277-2d.jpg'>

Is the Markov chain irreducible? If the Markov chain starts at state what is the probability that it is at state 3 after two
units of time? 10
3.(a)Give an example to show that a sufficient statistic need not be complete. 10
3.(b)Let X1,X2, ..., Xn be Li.d. Poisson (lamda) random variables. Suppose the prior distribution of
lamda is Gamma (alpha,beta). What is its posterior distribution? 10
3.(c)Assume X1, X2, ..., Xn is a random sample from a Poisson (lamda) distribution. Examine whether the sample mean is UMVUE for lamda 10
3.(d)Let T be an unbiased estimator for theta. Is it always true that T2 is NOT an unbiased estimator of theta 2 10

Let Xl, X2, ..., Xn be i.i.d. random variables from a N(mew,sigma2) population. Obtain the 1OO(1-alpha)% confidence interval
mew when sigma2 is known
for sigma2 when mew is known
Suppose X1,X2, ..., Xn are i.i.d. random variables following the uniform distribution on the interval theta, theta+1),
-infinity<theta<infinity.
Show that where min {X1...,Xn} and =max{X1,···,Xn is a minimal sufficient statistic. 10
4.(c)The lifetime (in hours) of a sample of size 6 each of two different brands of batteries are given in the table below:
<img src='./qimages/1277-4c.jpg'>
Using the Kolmogorov-Smirnov test statistic, examine whether the brands are different with respect to their lifetime distribution.
10
Assume that the population follow a Poisson law with parameter lamda, derive the SPRT for testing H0:lamda lamda0
against H1:lamda lamda1 (lamda1 lamda0).
SECTION
5. Answer the following:
5.(a)Consider a Gauss-Markov linear model theta, sigma2I), where B=(theta1, theta2, theta3)' and A
and theta1 theta2 theta3, theta1 theta3, theta Obtain a necessary and sufficient condition
for estimability of the linear parametric function I1 theta1 I2 theta2 I3 theta3 In the case of estimability,
obtain the BLUE and also the variance of the BLUE. 8
5.(b)Define Hotelling's T2 statistic. Discuss in detail, the applications of the statistic in various testing problems. 8
5.(c)Describe the linear systematic sampling procedure. Obtain the variance of the sample mean in the presence of linear
trend in a population of size n x where n and k are positive integers. 8
5.(d)Two independent random samples of sizes nl and n2 are drawn from two independent normal populations,
N(mew1,sigma2) and N(mew2,sigma2> where sigma2 is common but unknown. Explain how would you construct a 95% confidence interval
for the ratio mew1/mew2(assume mew2 not equal to 8

5.(e)Write short notes on the following:
5.(e)(i)Discriminant Analysis
5.(e)(ii)Confounding in factorial experiments 8

6.(a)in a Gauss-Markov linear model, show that any solution to normal equations, minimizes the error sum of squares.
illustrate the result through a small numerical example. 10
6.(b)In cluster sampling, obtain an unbiased estimator of the population mean when the clusters (of unequal sizes) are selected using SRS.
Compute also the variance of the estimator. 10
6.(c)In two stage sampling, obtain an unbiased estimator of the population mean if SRSWR is used at both stages.
Also derive the variance of the estimator. 10
6.(d)Using missing plot technique, estimate the missing observation in a RBD. Carry out the analysis by including the
estimated observation. 10

7.(a)Derive the characteristic function of a p-variate normal random vector. Hence obtain the first two moments of the random vector. 10
7.(b)Define stratified sampling. When is it useful? Obtain the sizes of samples from various strata under optimum allocation.
Find the corresponding variance of the sample mean. 10
7.(c)Discuss when the ratio estimator of the population mean under SRS is preferred to the conventional unbiased estimator.
7.(d)What is a factorial design? Briefly sketch the analysis of a 23 factorial design. 10

8.(a)What are principal components? Mention its uses. Explain how the principal components can be extracted from a dispersion matrix. 10
8.(b)If clusters are selected using SRSWR from a population of K clusters of size M each, then obtain the variance of an unbiased
estimator of the population mean in terms of the population intraclass correlation coefficient. 10
8.(c)Define a BIBD. Give a layout of such a design and outline its analysis. 10
8.(d)Write short notes on the following:
stage sampling and two phase sampling -their merits and demerits.
8.(d)(ii)Hotelling's T2 and Mahalanobis D2 ­their connections and applications. 10