Section
1. A tyre manufacturing company has three production units B and C which produce respectively 30% and 20% of the entire production. Unit A is known to produce defectives, unit B produces defectives and unit C produces defectives. What percentage of items in the entire produce is defective If an item is found to be defective, what is the probability that it was produced at unit 12
A bivariate random vector is specified by
P(X>x.
theta 1
Obtain the p.d.f. the marginal p.d.f's and the conditional p.d.f's. When will X and Y be independent 12
Examine whether the central limit theorem holds for the sequence Xn n of random
variables with
P(Xn P(Xn 1/2n3
P(Xn n 1.


Assume that X follow the Binomial, theta), and that the prior distribution of theta is the Unifonn, U I). Find the Bayes' estimator for theta using the squared error loss function.

X2 ....... Xn is a random sample from the
normal distribution with O mean and variance (sigma)2.
Examine the validity of the above statement.

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iii) T is sufficient for (sigma)2.
iv) T is the UMVUE for (sigma)2.

2. Define exponential family of distributions. Identify two distributions each which belongs to the family and does not belong to the family. Substantiate your answer.

For a continuous non-negative random variable X with E show that for t 0. src='./qimages/205-2b.jpg'> where P(X x). Evaluate the conditional expectation for the exponential model specified by x 0.

Show that for a sequence of random variables convergence in probability implies convergence in distribution. Is the converse true Substantiate your answer.

3. Based on a random sample of size n from the discrete Uniform distribution

3 ..... n
0 otherwise,
obtain the UMVUE of N. Is this estimator consistent
When will the Neyman-Pearson test and likelihood ratio test be the same? Obtain the likelihood ratio test for testing H0 p Po against Hl P not equal to p0 based on a random sample from the binomial, p).
A random sample of size n is taken from the Poisson distribution. Let N be the number of observations which are equal to 0. Obtain an unbiased estimator for and improve the estimator using Rao-Blackwell theorem.

4. Comment on the validity of the statement "the S.P.R.T. terminates with probability one". For the geometric distribution with parameter derive the S.P.R.T. for testing H0 p Po against H1 p p1 • Also obtain the O.C. function.
Compare the Chi-square test and Kolmogrov-Smirnov test as tests for goodness of fit. Show that the Kolmogrov-Smimov one sample test is distribution free. Also state the large sample distribution of the Kolmogrov-Smimov test statistic.
Describe the relationship between confidence interval estimation and testing of hypothesis. Let X1, X2, ......••, Xn be a random sample from (sigma)2 µ known. Obtain a confidence interval for (sigma)2.

Section
5 <img src='./qimages/205-5a.jpg'>
Find p such that 1 X2 X3) and (X1 X2 are independent.

Y1, Y2, Y3 and Y4 are independent stochastic variables with common variance a2 and Y2 and
Examine whether (alfa)1 is estimable.
Find the best unbiased estimator of
Obtain the variance of the best unbiased estimator of (alfa)2).

If r12, r13 and r23 are the simple correlation coefficients between X1 and X2; X1 and X3; X2 and X3 respectively then show that
src='./qimages/205-5c.jpg'>

Also examine the validity of the statement "the range of the panial regression coefficient is to +(lamda)."
Distinguish between simple random sampling with and without replacement. Write down the expression for the number of samples and the probability that a specified individual is included in the sample under both the sampling schemes SRSWR and SRSWOR.


Derive the normal equations for estimating the parameters of a linear regression model by the method of least squares.

6. Suppose that X follow the p variate normal distribution with co-variance matrix
src='./qimages/205-6a.jpg'>

Obtain the expression for the multiple correlation coefficient R 1.23 ... p·
Write down the model used for one way analysis of variance. The lifetimes (in hours) of samples taken from three different brands of batteries gave the following
src='./qimages/205-6b.jpg'>

Test the hypothesis that there is no difference in lifetimes for the three brands. Also write down the ANOVA (F2. 12, ·05 =3·89).

Suppose X (X1 X2, X3) 1 has co-variance matrix
src='./qimages/205-6c.jpg'>
Obtain the first two principal components of X.

7. Let N n 2 and 'the possible distinct samples are s1 2). s2 and s3 3). If SRSWOR is adopted, write down for i 3. If
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examine whether t is unbiased for the population mean . Also find the expression for if unbiased.

Bring out the relation between Hotelling's T2 and Mahalanobis D2. Explain how T2 can he interpreted as an extension of the student's t statistic. Describe che utility of Fisher's discriminent function in classification problems.

Discuss on the advantages of using a BIBO. For a BIBO with parameters show that
r(k and b
Also examine whether a BIBO is possible with
v b 34. r k 12 and 4.

8. What are inclusion probabilities Use this concept to define the Horvitz-Thomson estimator. If (alfa)k and (alfa)kl denote the first and second order inclusion probabilities under an arbitary ordered design of fixed size, show that
src='./qimages/205-8a.jpg'>

where v ts the expected effective sample size.

If Vran, Vprop and Vopt respectively denote the variance of the estimator of the population mean under SRSWOR, stratified sampling under proportional allocation and Neyman allocation. show that

Vran Vprop Vopt


In a 33 factorial experiment, each replication is divided into 3 blocks of size 9 each and the effect ABC2 is totally confounded. Explain the analysis of the experiment and set up the ANOVA table.