STATISTICS



Time Allowed: 3 Hours Maximum Marks: 300
Caniliilates shouldaJtempt Questions 1 and 5 which arecompulsory, andanythreeoftheremaining
questions selecting oJ least one question/rom e(f£h Section.
Assume suitahle data ifconsiderednecessary andinilicnte the same clearly.
Notations andsymbols used are as usual

SEcnONA
(Probability and Statistical Inference)
Answer any five parts
unbiased Coin is tossed three times Write down the sample space of this experiment. Define a random variable on this sample space and also its probability distribution.
(b)The random variables are distributed independently with p.d.f <img src='./qimages/1113-1b.jpg'> Determine the probability p[min (X1
(c)Find the characteristic function of Y aX b where X has the p.i.f <img src='./qimages/1113-1c.jpg'>


Given a random sample of n observation from a POisson distribution lamda), derive an estimate of lamda based on the frequency of x 0 and work out its asymptotic variance.
Explain the terms prior and posterior distributions. How ate prior distributions selected? What is the role of conjugate prior distribution?
Indicate the use of Wald-Wolfowitz run test and state the test-statistic along with its mean and variance under the null hypothesis.
X follows an exponential distribution with parameter k Prove the folloWing relations:
x1)p(X



(b)State and prove Chebyshev's in equality. The random variable X takes values a WIth probabilities 1/8 respectively Compute sigma and compare it with Chebyshev's bound.

Xl,X2....Xn are random samples from a uniform distribution over Find the p.d.f of U (X1,X2...,Xn)l/n

Let Xl, X2....,Xn be i.i.d, rectangular (theta-1/2,theta)+1/2random variables Is sufficient for theta? Is It complete? Give reasons for your answer
Given a random sample from NN(mew,sigma2), where both mew and sigma are unknown, find the informly most accurate unbiased confidence interval sigma2

(c)Show that If T1 and T2 are UMVUE's of their expectations, then T1,T2 lS UMVUE of E

Let Xl,X2...,Xn be i.i.d. random variables With p d.f having median theta Show that for testing H0:theta 0 against alternatives theta the Sign test is uniformly most powerful

(b)Let Let (xi yi) i 2..n be a random sample from a bivariate normal population with parameters (mew1,mew2, sigma1,sigma2,row) Find a UMPU test for pHo: row=row0 vs row not equal to row0
(c)Briefly explain Wald's SPRT Also state its merits and disadvantages.


SEcnON B
(Linear Inference, Multivariate Analysis, Sampling Theocy and Design of Experiments) Answer any five parts
(a)What is multicollinearity? Give the methods of overcoming this problem.

(b)Describe Tukey's test for non-addivity for a two-way layout with one observation per cell.

Derive the Shapiro-Wilk statistic for testing normality of errors in a regression model.

(d)Explain Lahiri's method for selection of a PPS sample

(e)Describe a method for detecting heteroscedasticity In a model

(f)Give the ANOVA table for a 24 experiment blocks of 8 plots each, such that no main effect or first order interaction is confounded
6.(a)ConSider the model
<img src='./qimages/1113-6a.jpg'> with the usual assumptions Determine a so that the BLUE's of beta1 and beta2 are uncorrelated.
(b)Explain the method of generalized least squares for estimating the parameters of a linear model, stating the assumptions underlying it.

In order to test the equality of mean vectors of N(mew i,sigma)i=1,2 define Hotelling's T2 statistic and sketch the derivation of its null distribution
Discuss how one can unbiasedly estimate the population variance using stratified Simple random sampling

(b)Let pie i and pie j denote respectively the inclusiOn probability of population unit i and the JOint inclusiOn probability of two distinct units i and j in a PPSWOR sample of Size 2. Express pie i and pie ij in terms of selection probabilities and verify that sigma N to 1piei=2 and that piei piej>=pieij where N is the population Size.
(c)Briefly explain the idea behind Randomized Response Technique and indicate its possible uses.

8.(a)Explain the concept of balance in the context of a confounded factorial experiment.

Define an m-ple lattice deSign with its parameters and giVe its analySiS.

(c)What are 'mixed effects' models? Indicate the analysis of two-way clasSified data with equal number of observations per cell, uSing a miXed-effects model.