Indian Forest Service Examination-2012

STATISTICS
Paper-I
Time Allowed Three Hours Maximum Marks 200
INSTRUCTIONS
Candidates should attempt Question Nos. 1 and 5
which .are compulsory, and any THREE of the
remaining questions, selecting at least ONE question
from each Section.
All questions carry equal marks.
Marks allotted to parts of a question are
indicated against each.
Answers must be written in ENGLISH only.
Assume suitable data, if considered necessary,
and indicate the same clearly.
(Notations and symbo. ls are as usual)
Important Note
All parts/ sub-parts of a question being attempted are
to. be answered contiguously on the answer-book. That
is, where a question is being attempted, all its
constituent parts/sub-parts must be answered before
moving on to the next question.
Pages left blank, if any, in the answer-book(s) must be
clearly struck out. Answers that follow pages left
blank may not be given credit.
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1. Answer the following.: 8x5=40
A bag contains 10 coins-one worth
10 rupees and each of other nine worth
5 rupees. A person draws one coin at a
time without replacement until he
draws the coin worth 10 rupees. wli:at
is the expected total amount he has
drawn (including the last coin worth
10 rupees)?
What is the distribution of a random
variable with characteristic function
1 -lt2 1 2
2 2
There are two coins one with probability
1 of heads and the other with
probability of heads. One of the coins
is picked at random and let Sn be the
number of heads obtained when the
chosen coin is. t ossed n times. Which of
the following is fare correct?
Given any E 0


limP(
lim
Sn 1
n 2
Explain your answer.
2
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If X has a Poisson distribution and the
prior distribution of its parameter A is a
gamma distribution with parameters
a and then show that the posterior
distribution of A given X x is a gamma
distribution with the parameters x
and 1 (J3 1). Also obtain the mean of
the posterior distribution of A.
State and prove Wald's fundamental
identity on sequential
analysis.
Prove that the SPRT terminates
with probability 1.
2. Let X be a random variable with normal
distribution 1). Find the distri-
..
bution of Y =ex. 10
State and prove Lindeberg-Levy central
limit theorem.
A fair coin is tossed independently n
times. Let Sn be the number of heads
obtained. Use Chebyshev's inequality to
find a lower bound of the probability
that
8
n differs from .!. by less than 0·1'
n 2
10
when n 100 and n 100000. 10
A monkey jumps to get a fruit from a
tree. Suppose the probability that the
monkey fails to get the fruit at the
nth attempt is Using Borel-Cantelli
n
lemma, show that the monkey will eventually
get the fruit with probability 1. 10
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3. LetXI, x 2 ...
of size n from
given by

Xn be a random sample
the uniform population
for . 2 2
elsewhere
Find .the maximum likelihood estimator
for e. Comment on this estimator. 10
State factorization theorem of sufficient
statistic.
Let ... be a random
sample from a distribution with p.d.f.

9

0 otherwise
where 0 8 Find a sufficient
statistic Jar e. 10
If XI, X 2 ..• Xn constitute a random
sample from the population given by
e x

O
0 elsewhere
show that the smallest sample value
1s a consistent estimator of the
parameter e. 10
Find the critical region of the likelihood
ratio test for testing the null hypothesis
H 0 Jl Jlo against the composite
alternative HI: Jl Jlo, on the basis of a
random sample of size n from a normal
population with known variance cr 2 • 10
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4. Describe the method of constructing
similar reg10ns m testing composite
hypothesis. Show that every test based
on a similar region will have NeymanPearson
structure with respect to a
sufficient statistic if the family of
distributions of T for Be 0 is
boundedly complete. 10
State and prove inversion theorem of
characteristic function. 12
On a genetic inbreeding experiment
with single pair of allelomorphs A.and
the progeny of a mating Aa x Aa results
m 50% heterozygotes Aa, where the
proportion of heterozygotes being
reduced by one-half at each generation.
Find the transition matrix of various
genotypes and the proportion of
heterozygotes at rth generation. 8
Write notes on the following Sx2=10
Kolmogorov-Smirnov two-sample
test
Wilcoxon-Mann-Whitney test
Section-B.
5. Answer the following 8X5=4Q
Describe the technique of analysis of
variance for testing equality of a number
of regression equations.
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Define 'estimable parametric function'.
State the conditions of estimability.
Show that such a function has a unique
linear estimator which is best unbiased.
Defme Hotelling's T 2 and Mahalanobis's .
D 2 statistics, and obtain the relationship
between them. Explain the uses of
these statistics.
Show that Horvit2-Thompson estimate
is an unbiased estimate of the
population mean. Obtain the variance of
the estimate of the mean under Horvit2-
Thompson estimate.
Prove that the number of treatments
common between any two blocks in a
symmetrical BIBD is
6. State the Cramer-Rao lower bound. Give
two examples, one in which the lower
bound is attained and another where it
is not attained.
What is principal component analysis?
Discuss its uses with examples. Show
that the principal components are all
uncorrelated.
Describe the AOV for a two-way
classification (random effects model)
10
10
with one observation per cell. 10
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. Describe Fisher's method of discrimination
between two groups on the basis
of multiple measurements. Explain how
this procedure can be represented as a
problem in regression. 10
7. Show that, for estimation of population
mean
A population consists of nk p units,
where k are positive integers
k). A systematic sample is selected
with k as the class interval. Show that
the sample mean is a biased estimate
of the population mean. Obtain an
unbiased estimate of the population
mean.
Compare ratio and regression· methods
of estimation. Show that the ratio
method gives a biased estimate of the
population total. Derive an approximate
10
10
expression for the bias. 10
Why is it said that a split-plot design
confounds main effects? Give an outline
of analysis of a split-plot design. 10
8. The yield of one plot of a p x p LSD
is missing. Estimate the missing
observation and give an outline of the
analysis of the design. 10
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For ·two-stage sampling, where the
first-stage units are of equal size, obtain
an estimator of population mean. Also
obtain the expression for the variance of

the estimator. 10
Discuss, in detail, how total and partial ..
confounding can be done in 3 3 -factorial
experiment. Give the partition' of d.f.,
contents of the blocks and calculation of
sum of squares in each case. 10
Write explanatory notes on the
following Sx2= 10
Lattice designs .
· ·Non-sampling errors

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