R.B.I.S.B. R.O. DSIM) P.Y.2013
PAPER II DESCRIPTIVE TYPE ON STATISTICS
(Maximum Marks 100) (Duration 3 Hours)

Instructions

The question paper consists of six sections. The candidate may attempt any five questions selecting not more than two from any section. In case the candidate answers more than five questions, only the first five questions in the chronological order of question numbers answered will be evaluated and the rest of the answers ignored.

QUESTIONS FROM EACH SECTION SHOULD BE ANSWERED ON SEPARATE ANSWER-SCRIPT/SUPPLEMENTS. In other words, a candidate may require/use minimum 2 to 4 supplements, in addition to Answer script.

Supplement should be attached to the answer script, before returning.

Each question carries 20 marks.

Answers must be written either in English or in Hindi. However, all the questions should be answered in one language only. Answer-books written partly in English and partly in Hindi will not be evaluated.

Each question should be answered on new page and the question number must be written on the top in left margin.

The answers of parts of the same question, if any, should be written together. In other words, the answer of another questions should not be written in-between the Parts of a question.

The Name, Roll No. and other entries should be written in the answer-scripts at the specified places only and these should not be written anywhere else in the answer script and supplements.

Candidate should use only Blue or Black ink pen/ballpoint pen to write the answers.

No reference books, Text books, Mathematical tables, Engineering tables, other instruments or communication devices (including cellphones) will be supplied or allowed to be used or even allowed to be kept with the candidates. Violation of this rule may lead to penalties. Use of non-programmable electronic calculator is permitted.

ALL ROUGH WORK MUST BE DONE IN THE LAST THREE OR FOUR PAGES OF THE ANSWER SCRIPT.

Answers will be evaluated on the basis of logic, brevity and clarity in exposition.

Marks will be deducted for illegible hand-writing.
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Probability and Sampling
1.

In a SRSWOR of n clusters from a population of N clusters each containing M elements, obtain an unbiased estimator of Y (population mean per element . Also obtain its variance. State the unbiased estimator of the variance obtained.

The following data relates to a random sample of 5 clusters from a population of 25 clusters each consisting of 2 consecutive plots. The value of y are the areas of sample paddy fields.
Cluster
1
2
3
4
5
Area
24
16
12
6
25
21
16
18
3
3
Estimate the average area under paddy per field together with its variance.
2.

State the strong law of large numbers (SLLN). Show that every sequence {}nXof independent random variables with uniformly bounded variances obeys SLLN.

Examine whether the strong law of large numbers hold for the sequence independent and identically distributed random variables with a common probability density function.
()???≥=−otherwisexxxf0178
3.

Define Convergence in probability
Convergence in rth mean
almost sure convergence.
If XXpn?→? and g is continuous function show that

is a sequence of iid random variables with a common density function

Show that 45?→?pnX where nX is the sample mean.
Linear Models and Economic Statistics
4.

State the assumptions that must be satisfied in multiple regression. Briefly state how to check violation of assumptions.
12

Assume regression model and n observations.
Prove that regression sum of squares ()221ˆxxi−Σβ
Data on 10 observations gives: ()267.3ˆ2=−Σiiyy, 3.63=Σiy 49.4232=Σiy 139=Σix,22392=Σix
Estimate parameters 0β and 1βassuming positive correlation between x and y. Find ()0ˆvarβ and coefficient of determination.
5.

Describe one way analysis of variance model for testing equality of k treatment effects, there being in observations on the thi treatment; i State the assumptions and obtain testing procedure. Write ANOVA table.

A random sample of 16 observations was selected from each of four populations. Part of ANOVA table is given below.
Source of
Variation
d.f.
Sum of squares
Mean sum of squares
F
Treatments



80
Error



Total

1500
Value of F at level of significance with appropriate d.f. =2.76.
Fill in the missing values and complete the table.
What is your conclusion at level of significance?
6.

What do you mean by an index number? What purpose does it
serve?
What are the requirements of time reversal and factor reversal tests? Show that Fisher's ideal index number satisfies both the tests.

Using the following data compute Fisher's price index number and verify that it satisfies both the tests mentioned in
Commodity
Base year
Current year
Price
Quantity
Price
Quantity
P
5
60
8
60
Q
2
100
2
120
R
5
50
6
70
S
8
40
10
30
Statistical Inference
7

Let 0T be the UMVUE of and V be the unbiased estimator of 0. Then prove that 0T is UMVUE if and only if 0)(0=TVEΘ∈∀θ. Assume that second moment exist for all unbiased estimators of and
13

Let nXXX,...,,21be iid rvs with μand σ are unknown.
Find the UMVUE of where k is some real number.
Let nXXX,...,,21be iid rvs with discrete uniform.

Find UMVUE of Ne
8.

Let nXXX,...,,21are iid rvs with Θ∈θ. Find MLE of θ.
Let

Find the mle of θ from the following data
1.

Let nXXX,...,,21be iid rvs with Obtain Bhattacharya bound for
9.

Let 1,...,21kXXXand 2,...,,21kYYYare iid rvs
with and respectively, where 1n and 2nare known. Find Neyman Structure test for testing 210:ppH= against 211:ppH>.

A random sample of 10 households selected from two cities (Mumbai and Delhi), revealed that their weekly medical expenses were as under:
Mumbai

20
16
10
9
18
21
8
11
27
24
Delhi

12
19
23
17
26
22
15
13
14
25
Using the Wald-Wolfowitz run test, verify at 05.0=α, if the two population have the same underlying distributions.

Stochastic Processes
10

A fair coin is tossed repeatedly with results ,...,,210YYY that are 0 or 1 with probability half each. For 1≥n let 1−+=nnnYYX be the number of one's in the thn)1(−and thntosses. Find the expression for each of the probabilities.

Whether nX has a Markov Property? Justify.
14

A transition probability matrix P is said to be doubly stochastic if the
sum over each column equals one. If such a Markov Chain is
irreducible, aperiodic and consists of m+1 states show that limiting
probabilities are given by 11+m .
Let nNbe the number of heads observed in the first n flips of a fair
coin and nXequal to nNmod find ∞→nlimP nN is multiple of
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Two gamblers A and B bet on successive independent tosses of a coin that lands heads up with probability p. If the coin turns up heads gambler A wins a Rupee from Gambler B and if the coin turns up tails gambler B wins a Rupee from Gambler A. Thus the total number of Rupees among the two gambler stays fixed say N. The game stops as soon as either gambler is ruined. Describe this as a Markov Chain and found its stationary distribution. Is it the limit distribution?

Consider the points marked on a straight line. Let nX be the Markov Chain that moves to the right with probability 32 and to the left with probability 31 and subject to the condition that if nXtries to go the left from 1 or to the right from 4 it stays there. Find the limiting amount of time the chain spends at each point.
12
A time series model is specified by tttteYaYaY+−=−−2212.
where teis a white noise process with variance 2σ.
Determine the values of a for which the process is stationary.
Derive the auto covariances kυfor 2≥k
Show that the autocovariance function can be written in the
form kkkaBkAa+=υ
for some values of A and B which you should specify in terms of the constants aand 2σ. Hence show that autocorrelation kρ

Give a testing procedure for testing whether underlying
Time series can be modeled as process given above.
Multivariate Analysis
13

Describe an orthogonal factor model with all its assumption and explain the following terms Factor loading specific variance communality.
Explain in brief what is factor rotation.

For the following covariance matrix obtain the loading matrix L and specific variances assuming one factor model and using principal component method.

Obtain the proportion of total variance explained by first factor.
15
14

Define canonical correlations and the canonical variates. Show that the first canonical correlation is larger than the absolute value of any entry in the matrix .12ρ Where

is a correlation matrix of vector.

Obtain first canonical correlation and its associated canonical variate pair for the following partitioned correlation matrix.

15

Let 21,pp denote the prior probabilities of two multivariate populations and and )(),(21xfxfdenote their probability density functions. Obtain the classification rule that minimizes total probability of misclassification. Hence derive minimum TPM rule when ifis p-variate normal with mean vector common variance matrix Σ.

Suppose that 301=n and 222=n observations are made on two random vectors 1X and 2X which are assumed to have bivariate normal distribution with a common covariance matrix Σ but possibly different mean vectors and 2μ. The sample mean vectors and inverse of a pooled covariance matrix are as follows:




?????−−=−147.108423.90423.90158.1310262.002483.0,039.00065.0121pSXX
Test the hypothesis against
Obtain the Mahalanobis sample distance .2D
(iii)Obtain the minimum TPM rule for classification.
Classify the observation ()044.021.0−−=′X to or 2π.
Numerical Analysis and Basic Computer Techniques
16.

If is the approximation to the solution X of the system of linear equations AX then show that the next approximation of X using Gauss-Seidel iteration method is obtained by where L)−1 b − and D the diagonal part, L the lower triangular part of A.
…n.
16
Using this method obtain for the following system of equations
2x1− x2
−x1+ 2x2 − x3 1
− x2 +2x3 1
Take initial approximation as 0 0]′

Derive the Newton-Cotes Quadrature formula
∫Σ==ban0kkk)x(fdx)x(fλ
and deduce the Simpson's rule in the form

using method of interpolation. Hence evaluate the following integral ∫102)cos(dxxx
17

Define forward difference operator Δ and prove that
Evaluate the following, interval of differencing being unity:

Δ2E3x3

The following values of the function for values of x are given as =4. Find the value of and also the value of x for which is maximum or minimum using Lagrange's interpolation formula.
18

Answer the following questions:
Explain TCP/IP reference model in networking.
What is a link (transmission media)? Explain point-to-point link and
broadcast link.

Answer the following questions:
What do you understand by a relation in Relational Database Management System? Explain how to select specific attribute columns and tuples from a relation.
Explain Binary search technique for the following numbers by taking search . key as 40:
12, 18, 24, 30, 35, 38, 40, 45, 50.
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