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lndJan forest Examination
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r:Y7 1. I D-VSF-L-FGA I
STATISTICS
Paper I
I Time Allowed Three Hours I I Maximum Marks: 200]
INSTRUCTIONS
Candidates should attempt Questions No. 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 symbols are as usual)
SECTION A
1. Answer any four parts 4XJ0=40
(a)Box I contains 4 white and 2 red balls, and
Box II contains 3 white and 5 red balls. Two
balls are chosen at random from Box I without
observing their colours and are put in Box II. A
ball is then picked from Box II. What is the
probability that it is white

certain manufacturing plant uses a specific
bulk product. The amount of product used in a
day can be modelled by an exponential
distribution with parameter 4 (measured in
tons). What is the probability that the plant win
use more than 4 tons on a given day
(c)People either like dogs or dislike them. If a
statistician wants to estimate the probability p
that a person likes dogs, how many people must
be included in the sample Assume that the
statistician will be satisfied if the error of
estimation is less than 0.04 with probability
equal to 0.90. Assume also that the statistician
expects p to lie in the neighbourhood of 0.06.
[You may need some values from the ones given
below
z0.025 1.96, z0·05 1·65]
(d)Let X1...,X10 be a sample from a Bernoulli
distribution with p for some
p belongs Find the uniformly minimum variance
unbiased estimator (UMVUE) of p2:
have 5 independent normal distributions
with unknown means mew1....mew5 respectively and
an unknown common variance sigma2. From each of
these distributions a sample of size 10 is drawn;
thus, Xi,1....Xi,10 denotes the sample of size 10
from N(mew1, sigma2) Obtain an unbiased estimator of
sigma2 What is the distribution of this estimator

Let A,B and C be three independent events with
0·2, P(AcnBcncc) 0.42,
P(A n B n 0·015.
Let p P(C nAcnBc). Show that p equals
either 0·14 or 0·18.
[Here Ac, Bc and cc denote the complements of
the events and C respectively.] 10
(b)What is the characteristic function of a random
variable X whose probability density function is
<img src='./qimages/1487-2b.jpg'>

be a random vector with joint density
given by
<img src='./qimages/1487-2c.jpg'>

Evaluate c.Show that X and Y are uncorrelated.
Are X and Y independent?
(d)Let X1,X2,....be random variables. Give an
example for each of the following:
Xn converges in probability to but Xn
does not converge almost surely to X.
(ii)Xn converges in probability to but
does not converge for any p>0. 10

3.(a)Let X be a Poisson random variable with
unknown mean lamda Find a function of lamda for which
the amount of information in a sample of size n is
independent of lamda. 10

(b)Let Y1, Y2 ... Yn be a random sample with
having a density function
<img src='./qimages/1487-3b.jpg'>

Show that the mean of the sample, is a
sufficient statistic for alpha. State clearly the theorem
you use.
(c)Let theta be a statistic that is normally distributed
with an expected value and a variance equal to theta
and sigma2 theta respectively. Explain how to obtain a
(1-alpha)100% confidence interval for theta. 10

Let mew1 and mew2 be the average lifespan of a
rhinoceros in captivity and in the wild
respectively. The data of the lifespan of
9 rhinoceros in captivity and 9 rhinoceros in
the wild were recorded as y11, y12,... ,y19 and as
Y21,y22,... ,y29 respectively. The data showed
<img src='./qimages/1487-3d.jpg'>

At 0·05 level of significance decide whether the
lifespans of the rhinoceros in captivity and in the
wild are same or not. 10
[You may need some values from the ones given
below
at degrees of freedom 16, t0.025 2·120,
t0·05 1·746
at degrees of freedom 18, t0.025 2·101,
t0·05 1·734
z0·025 1·96, z0·05 1·65]

4.(a)Suppose that yij alpha i beta tij Eij, i 1,...n,
j=1,.... where alpha i and beta are unknown
parameters, tij are known constants, and Eij are
i.i.d. random variables with 0 mean. Find
explicit forms for the least square estimators
of beta, alphai, i 1,... n.
(b)What is a randomized block design Explain
how to obtain the confidence interval for the
difference between a pair of means in a
randomized block design.

(c)Use the method of least squares to fit a straight
line to the data given below.
X y
0
0
0 1
1 1
Sketch the line.
What are the variances of the estimated slope
and intercept Are they correlated
(d)Let p be the probability that a randomly
chosen Indian is a vegetarian. We want to test
H0:p 0·2 against H1:p 0·6 on the basis of
a sample of size 10. Let Y be the number of
people in our sample who are vegetarians. If we
have a rejection region then
(i)what is the Type I error
(ii)what is the Type II error

SECTION B
5. Answer any four parts
(a)Let X be a random variable with mean 11 and
variance 9. Using Chebychev's theorem find the
following
lower bound for P(6 X
(ii)the value of c such that
0·09.
(b)Suppose X1 and X2 are two normal random
variables both with mean but with variances
1 and 16 respectively. On the same graph sketch
both the probability density functions and on
another graph sketch both the probability
distribution functions, clearly labelling the
curves. If a and b are such that
then which of the
three is correct: a>b or a=b
(c)Two independent random samples of sizes m
and n are taken from two independent normally
distributed populations with unknown variances
sigma1 2 and sigma2 2. Explain how you would construct a
90% confidence interval for the ratio sigma1 2/sigma2 2.

(d)The number of students in four courses in two
different universities A and B are given in the
table below.
Course A B
I 27 32
II 31 29
III 26 35
IV 25 28
With this data explain how to test the null
hypothesis (at a level of significance that the
population relative frequency distributions in
the courses at universities A and B are identical
against the alternate hypothesis that the
population relative frequency distributions are
shifted in respect to their relative locations.
(e)Consider the randomized block design with v
treatments, each replicated r times. Let ti be
the treatment effect of the i-th treatment. Find
<img src='./qimages/1487-5e.jpg'>
summation miti are the best linear unbiased estimators of summation liti and summation miti respectively
and summation li=summation mi=summation limi=0

6.(a)Let X be a random variable with 1/10
for 1,...,9 and let Y be a random variable
(independent of with a uniform distribution on

(i)What is the distribution function of X Y
(ii)What is the moment generating function of
(iii)What is the correlation function between X and 10
(b)Let Xl,X2....be a Markov chain with state space
and transition probability matrix
<img src='./qimages/1487-6b.jpg'>
What is the invariant distribution of this Markov
chain? 10
(c)(i)Obtain the distribution of the sum of 30
independent Poisson random variables,
each with mean 1/3.
(ii)Obtain the distribution of the sum of 30
independent Bernoulli random variables,
each with mean 1/3. 10
(d)State the Borel-Cantelli lemma and its converse. 10

7.(a)From a finite population we obtain a sample of
size n.
(i)Obtain the variance of the sample mean,
when the sampling is done with
replacement.
(ii)Obtain the variance of the sample mean,
when the sampling is done without
replacement.
(iii)If the population has mean mew and variance
sigma2, then what can you say about the
asymptotic distribution of the sample
mean?
bag has 5 balls some red in colour and the
remaining blue. You pick out 3 balls without
replacement and observe that 2 of them are red
and 1 of them is blue. What is the maximum
likelihood estimate of the number of red balls in
the bag? 10
(c)Explain the notions of the power of a test and the
P-value of a test. 10
(d)What is a hxk contingency table and how do you
test the hypothesis that the two classifications
producing the hxk contingency table are
independent of each other 10

8.(a)Given the points yn) of a scatter
diagram, let y a bx be the fitted regression
line. Describe a measure of scatter about the
regression line and its relation to the sample
variance of y and the sample correlation
coefficient of x and y. 10

(b)Let X1, X2,... be independent random variables
with a common probability density function f.
Compare the Neyman-Pearson and the
sequential probability ratio test methods of
testing H0:f f0 versus H1:f f1. What are the
errors of acceptance and rejection in both these
methods? 10
(c)The blood group classifications of human beings
are and AB with relative frequencies
proportional to r2, (p2 (q2 2qr) and 2pq
respectively, where p q r 1. In a large
sample of N persons tested, the observed
frequencies in the four types were found to be n1,
n2, n3 and n4 respectively. Find the estimates of
the proportions q and r and also the large
sample variances of the estimates.
(d)Suppose X1,X2,...,X20 are independent random
variables, each having a normal distribution with
known mean and unknown variance sigma2, Obtain
a lower bound for the variance of any unbiased
estimator of sigma2. State any theorem you use. 10