civils mains 2013 STATISTICS Paper I

Time allowed: Three Hours

Maximum Marks: 250
Question Paper Specific Instructions
Please read each ofthe following instructions carefully before attempting questions:
There are EIGHT questions divided in two SECTIONS and printed both in HINDI and in ENGLISH.
Candidate has to attempt FIVE questions in all.
Questions no. 1 and 5 are compulsory and out of the remaining, THREE are to be attempted choosing at least ONE from each section.
The number ofmarks carried by a question/part is indicated against it.
Answers must be written in the medium authorized in the Admission Certificate which must be stated clearly on the cover of this Question-cum-Answer Booklet in the space provided. No will be given for answers written in a medium other than the authorized one.
Assume suitable data, ifconsidered necessary, and indicate the same clearly.
Unless and otherwise indicated, symbols and notations carry their usual standard meaning.
Attempts of questions shall be counted in chronological order. Unless struck off, attempt of a question shall be counted even if attempted partly. Any page or portion of the page left blank in the answer book must be clearly struck off.

SECTION A

Ql. In the course of an experiment with a particular brand of DDT on flies, it is found that 80% are killed in the first application. Those which survive develop a resistance, so that the percentage of survivors killed in any later application is half that in the preceding application. Thus 40% of the survivors of the first application would succumb to the second, 20% of the survivors of the first two applications would succumb to the third, and so on. Find the probability that
a fly will survive four applications.
it will survive four applications, given that it has survived the first one.

The joint distribution of two random variables X and Y has the pdf
<img src='./qimages/215-1b-2c.jpg'>
Derive the marginal distribution of X and the conditional distribution of Y givenX. 10

Let Xl, X2 be independent random variables following Poisson distribution with parameter A.. Show that Xl Xz is sufficient for lambda Verify if Xl 2X2 is also sufficient for lambda 10

Let X be a negative Binomial random variable with parameters rand teta, r being known and teta unknown. Suppose the problem is to test Ho: teta 1/2 against HI teta 1/2. Find the UMP test for the problem and obtain its power function. 10

Use sign test statistic to find a non-trivial distribution free confidence interval for population median of a continuous distribution.10

Q2. If denotes the characteristic function (ch.f.) of a random variable with distribution function show that 1J is a characteristic function. Hence find its corresponding distribution function. 20

(b)For the Laplace distribution
<img src='./qimages/215-2b-2c.jpg'>
find the moment generating function. 15

Let be jointly distributed with probability density function
<img src='./qimages/215-2c-2c.jpg'>
Find Fisher information in a sample ofn pairs. 15

Q3. Using Rao Blackwell Theorem, derive the uniformly minimum variance unbiased estimator (UMVUE) of an estimable parametric function based on a complete sufficient statistic. Obtain the UMVUE of P[X r ... m where Xl,X2 ... Xn are independently identically distributed (i.i.d.) Bin(m, p). 20

For one parameter exponential family obtain UMPU test for testing
<img src='./qimages/215-3b-2c.jpg'>

Let X be distributed as Bin(n, and 1 p be the prior probability density of p. Then obtain the Bayes estimator (under squared error loss) and the Bayes risk. 15

Q4. Let N be the number of observations required by the sequential probability ratio test (SPRT) for testing
<img src='./qimages/215-3b-4a.jpg'>
expression of the percentage saving in sample sizes under Ho based on SPRT over the fixed sample size based MP test. Take the strength of each procedure as (alfa,beta). 20
State the strong law oflarge numbers. Decide whether the strong law of large numbers holds for the sequence of mutually independent random variables Xk with distribution 15
<img src='./qimages/215-3b-4b.jpg'>

To compare two brands of tyres, the following milages Cin miles) were obtained for 8 tyres of each kind:
BrandA: 32·1, 20·6, 17·8, 28·4,1%, 21·4, 19·9, 30·1
Brand 19·8,27·6,30·8,27·8,34'1,18'7,16·9,17·9
Test the null hypothesis at a =0·05 that the two samples come from the same population using Mann Whitney test.
<img src='./qimages/215-4c.jpg'>

SECTIONB

Q5. Consider the following linear model:
<img src='./qimages/215-5a.jpg'> 10

<img src='./qimages/215-5b.jpg'> 10

(C)Show that systematic sample is better than SRSWOR if Pw< where nand N aresample and population sizes n-1 w respectively and Pw is the intraclass correlation between pairs of units that are in the same systematic sample. 10

What do you understand by a connected block design? The following is the incidence matrix of a block design with four treatments and five blocks:
<img src='./qimages/215-5d.jpg'>
Compute the C-matrix of the design. 10

Define multiple correlation coefficient P1.23 ... p between <img src='./qimages/215-5e.jpg'> 10

Q6. Let C be the C-matrix of a block design with parameters and let Q be the vector of adjusted treatment totals. Show that, under the usual additive fIxed effects linear model C.. and Co2 where r is the vector of treatment effects and 0 2 is the common experimental error variance. Also show that is estimable if and only if lies in the row space of C. 20

Let Xl... XN be a random sample from <img src='./qimages/215-6b.jpg' where
<img src='./qimages/215-6b1.jpg'>

Suppose it is required to estimate the average value of output of a group of 5000 factories in a region so that the sample estimate lies within 10% of the true value with confidence coefficient of 95%. The population coefficient of variation is known to be 60%. Determine the equation to calculate minimum sample size. 15

Q7. Show that for a BIBD with v nk for a positive integer, b v r l. Show that for a symmetrical BlED with v even, r lambda. is a perfect square. 20

Define Mahalanobis measure of distance squared between two populations with a common positive definite dispersion matrix and show that it is a non decreasing function of the number of characteristics. 15

A rural block in a district was divided into 3 strata. The following table gives the number of villages and standard deviation of area under wheat crop for different strata:
<img src='./qimages/215-7c.jpg'>
A sample of 20 villages is to be drawn. How many villages should be selected from each strata under 15
proportional allocation?
Neyman allocation?

Q8. <img src='./qimages/215-8a.jpg'>

Let X has the correlation matrix R given by
<img src='./qimages/215-8b.jpg'>
Obtain the first two principal components and the population variance explained by the first two principal components. 15

Define a ratio estimator. Show that it is biased. Suggest a suitable sampling procedure for which the classical ratio estimator of a population ratio is unbiased. For the suggested procedure, propose an unbiased estimator of the variance of the ratio estimator. 15