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 sub-parts
For the three mutually exeluSive events the following probabilities are giVen PCA) 2/3 PCB) 1/4 pee) 1/6 Calculate peA v B v and comment about result
Define characteristic function State its properties. Examine whether the following relations are characteristic function . <img src='./qimages/1028-1b.jpg'>
(c)Slate the properties of the probability density function of a random variable X Examine whether the following is a probability density function Also evaluate P(2 x <img src='./qimages/1028-1c.jpg'> Where 0 is a constant
Let X be a normal variate With mean mew and variance d Estimate mew and d by using method of moments
Explaining the notations describe Chi -square good-ness of fittest
Suppose and X2 are identically and identically distributed POisson variates with parameter 0Show that the statistic T 3X, is not sufficient for 8.

State the conditions for the eXistence of weak law of large numbers. Examine whether weak law of large numbers holds for the mean of a sequence of independent variates X with <img src='./qimages/1028-2a.jpg'>
The Joint probability density function of two random variables X and Y is 1/8 0 2 Find Marginal distributions of X and Y

Let X be a random variable having standard Cauchy's distribution Obtain the probability denSIty function 0fX' and Identify the resulting distribution
Explain the concept 0f conSistency Show that for Cauchy's distribution the sample median is a conSistent estimator 0f the population mean

State Chapman-Rob bins inequality. Explain how it differs from Cramer-Ran inequality

Prove that If e is the maXImum like liho0d estimator 0f e and ¢ is a one-to one function of then is the maXimum likelihood estimator for

A sample of Size 1 is taken from probability density function, <img images/1086-4a.jpg'> Find most powerful test 0f Ho>0 =oo against at level alpha

(b)Discuss Wilcoxon-Mann-Whitney test and examine its conSistency

Discuss likelihood ratio test and discuss its conSistency



SECTION B
(Linear Inference, MuIti-variateAnaIysis, Sampling Theory and Design of Experiments) Answer any FIVE sub-parts
IfY X beta is a general linear hypotheSIs model of full rank. Find the estimate of the unknown parameter using appropriate method when the distribution of e is unspecified

Define regressiOn analySIs. Write the form of Simple linear regressiOn model State the various assumptions of the model

Define regresSIon estimator When it becomes conSistent? Write the reasons for becoming regressiOn estimator biased

Define cluster sampling Write the basic difference between the cluster sampling and stratified sampling

If Ddenotes a BIBD with parameters and denotes its complement"'Y deSign with parameters then show thaI, !..

Define total and partial confounding. In the following arrangement of 23 factorial deSlgns identify the confounded factorial effects·-
<img images/1086-5f.jpg'>
Let X be distributed N(mew, find the characteristic function 0f X

What are the situations where discriminate analySiS is used? Give some examples in support of your answer Write the objectives of two group discriminate analySiS Explain how the discriminate critenon
is calculated

Explain the concept of one-way clasSified data Write the linear model for it Obtain the least square estimate of the parameters involved is it

If finite population correction is ignored, then show that
<img images/1086-7a.jpg'>
where V ope V prop and V m denote variance of optimum allocation, allocation, proportional allocation and simple random sample respectively

Prove that the variance of the mean of systematic sample is

<img images/1086-7b.jpg'>
where k N and p is the intraclass correlation coefficient between the units of the same systematic sample
Explain the concept of ratio estimator Prove that 1n large sample, with Simple random sampling, the ratio estimate YR has a smaller variance than the population estimate Y obtained by Simple expansiOn, 1f
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Define symmetrical factorial experiment Obtain all possible main effects and intersections for factorial experiment Discuss Yates method 0f analySiS 0f 2n factorial experiment

Give the layout of a randomized block deSign conSldenng 6 treatments mto 5 blocks Whether this is a balanced deSign? Give the complete analySiS of variance of randomized block deSign with k treatments and r blocks. (Derivation of formulate not needed)

Define Simple lattice deSign. Construct a Simple lattice deSign taking 16 treatments Derive the complete analySiS of Simple lattice deSign