Time Allowed: 3 Hours Maximum Marks: 300
Caniliilates shouldaJtempt Questions 1 and 5 which arecompulsory, andanythreeojtheremaining
questions selecting oJ least one questionjrom e(f£h Section.
Assume suitahle data ifconsiderednecessary andindicntethe sameclearly.
Notations andsymbols used are as usual

SECTION A
(Probability and Statistical Inference)
Answer any five sub-p arts 0f tee fallowing
Given that are events with A and C mutually excluSive, B and C Independent and 1/4 P(AnB calculate
p(B bar intersection C bar)
(b)State Inversion theorem Find the distribution to which the characteristic function corresponds

(c)Independent trials consisting of rolling of two fair dice are performed What is the probability that an outcome of 5 appears before an outcome of where the outcome of aroll is the sum of the dice?
drug manufacturer wants to test the efficacy of a new medication. Denoting by theta the percentage of all the partients given the medication who recover from the disease he sets up H0: theta=0.90, H1:theta 60. His test statistic iS the observed number of successes In n 25 trials and he will accept the null hypothesis if Find the probability of type I and Type II errors denoted respectively by alpha and beta What happens If the acceptance regiOn is changed to x3 16?

E(T1)=theta= var(T1)=sigma1 2,var (T2)=sigma2 2 and T1 and T2 are Independent, then show that T=lamdaT1+ (1-lamda)T2 is an unbiased estimator of theta for which the Variance is minimized, when <img src='./qimages/1123-1e.jpg'>

(f)Explain run test The folloWing arrangement of defective and non-defective pieces produced In the given order by a certain machine was observed nnnn dddd nnnnnnnnnn dd nn dddd n dd nn Test for randomness at 0.01 level of Significance (Area under standard nornal curve from 0 to 2.575 is 0.495)

2.(a)For any integer-valued random variable show that <img src='./qimages/1123-2a.jpg'> where is the probability generating function of X

(b)Let Z be independent and uniformly distributed over(0,1) Compute

Define various modes of convergence of a sequence of random variables. Prove that convergence almost surely implies convergence in probability.

3.(a)Giving example,define sufficient statistics and state Fisher-Neyman criterion for the existence of sufficient statistics Let Xl,X2...,Xn be a random sample from the distribution that has p d. f <img src='./qimages/1123-3a.jpg'> Check whether Y1 K1+ X2+... +Xn is a sufficient statistic for theta
Describe Wilcoxon Signed-rank test and derive the approximate distribution of the test statistic.

Let Xl,X2...Xn be a random sample from N(theta,1) distribution, where theta is unknown Show that there is no uniformly most powerful test of the Simple hypotheSiS H0: theta theta', where theta' is a fixed number, against the alternative composite hypothesis H1: theta not equal to theta'

Discuss least squares method of estimation and the relation between least squares estimators and best linear unbiased estimators.
(b)Describe briefly the Wald's sequential probability ratio test. If for the choice A=1-beta/alpha, beta=beta/1-alpha the SPRT terminates with probability 1 and is of strength(alpha', then
prove that <img src='./qimages/1123-4b.jpg'>


(c)Let Y1<Y2<Y3 be the order statistics of a random sample of Size 3 from the uniform distribution having p.d.f 1/theta, x <theta,0<theta<infinity and zero elsewhere. Show that 4Y1 and 2Y2 are both unbiased stalistics for theta. Find the variances.
SEcnON D
(Lin ....r Infermce, F4ultivariate Analysis, Sampling Theory and Design of Experiments)
Answer any five sub-p arts 0 f the followmg
(a)Describing briefly the concept of multiple regressiOn, define multiple and partial correlations and obtain their formulae.

X is distributed as N5(mew,sigma), find the distribution of
x4

a Simple random sample of size 200 (without replacement) from a population of Size 1000, 75 persons happened to be smokers. Estimate the proportion of smokers and total number of smokers in the population and calculate 95% confidence interval for the proportion of smokers and also for the total number of smokers in the population (Giver that for 95% confidence probability, 1.96)
X intersection N sigma), prove that every linear function of components of X is univariate normal.

(e) Suppose that coefficient of variation of income of households In a region with 2000 households is 75%. How large a random sample of households is required for a margin of error ± with 95% a confidence coefficient7 (For 95% confidence probability, t 196)
What is a Latin square design? Give the analysis of an m x m Latin square design, with one missing value.

Discuss the procedure of clasSification of an observation into one of two known multivariate normal populations.

(b)Describe curvinear regression. Given that
<img src='./qimages/1123-6b.jpg'> fit the exponential curve y a3 applying the method of least squares.

X1 K2,...,Xn have a joint normal distribution, show that a necessary and sufficient condition for one sub set of random variables to be independent of the sub set consisting of the remaining variables is that each covariance of a variable from one set and a variable from the other set be zero.
Define ratio estimator in srswor show that the mse of R cap is approximately giVen by <img src='./qimages/1123-7a.jpg'>
(Retaining only the first tenn in an expansion)
a stratified population with two strata, the values ofWh, sh and Ch are as follows: <img src='./qimages/1123-7b.jpg'> If the cost function is of the form C1nl+c2n2, find the values of n1/n and n2/n that minimize C for a given value of V(y bar where the symbols have their usual meanings Also find the sample Sizes n1 and n2 for V(y bar ignoring f.p.c
double sampling for stratification, If the first sample is random of Size the second sample is a stratified random sub sample of the first of Size nk=vkn'k,where and Vh are fixed, then show that <img src='./qimages/1123-7c.jpg'> Where S2 is the population variance and other notations have their usual meanings.
8.(a)Discuss the two-way analysis 0f variance model giving its analysis and ANOVA table.

(b)Explain confounding in factorial experiments. Differentiate between complete and partial confounding. Set up a statistical model for a two factor factorial experiment RBD and give the ANOVA table with expected mean squares.
(c)WhatlS a BIBD? Write down the parameters of a BIBD, and state and prove the relations that serve as a necessary condition for the existence 0f a BIBD.