(Download) UPSC IAS Mains Exam Paper - 2017 : Statistics
STATISTICS
Paper - 1
Time Allowed : Three Hours
Maximum Marks : 250
QUESTION PAPER SPECIFIC INSTRUCTIONS
(Please read each of the following instructions carefully
before attempting questions)
There are EIGHT questions divided in two Sections and printed both in
HINDI and in ENGLISH.
Candidate has to attempt FIVE questions in all.
Question Nos. 1 and 5 are compulsory and out of the remaining,
THREE are to be attempted choosing at least ONE question from each
Section.
The number of marks carried by each question/part is indicated against it.
Answers must be written in the medium authorized in the Admission Certificate
which must be stated clearly on the cover of this Question-cum-Answer (QCA)
Booklet in the space provided. No marks will be given for answers written in a
medium other than the authorized one.
Assume suitable data, if considered necessary, and indicate the same clearly.
Unless and otherwise indicated, symbols and notations carry their usual
standard meanings.
Attempts of questions shall be counted in sequential order. Unless struck
off, attempt of a question shall be counted even if attempted partly. Any page
or portion of the page left blank in the Question-cum-Answer Booklet must be
clearly struck off
SECTION-A
Q1. (a) Ten percent of a certain population suffer from a serious
disease. Two independent tests are given to a person suspected of the disease.
Each test gives a correct diagnosis 90% of the time. Find the probability that
the person really suffers from the disease, given that,
(i) both tests are positive
(ii) only one test is positive.
(b) Let X be a continuous random variable with the probability density
function
Find the cumulative distribution function and the probability density
function of Y = x2.
(c) Let {Xn} be a sequence of independent and identically
distributed random variables with common probability function
Check whether {X} obeys the (it) strong law of large numbers and (i) weak law
of large numbers.
(d) Let x1, x2,..., Xn be n observations,
each independently uniformly distributed between 0-1/2 and 0+1/2 Find a general
form of maximum likelihood estimator (MLE) of 0. Verify that
are both MLEs, where Y1 = Min Xi and Yn, =
Max Xi.
(e) Let X1, X2, ..., Xn, be a random sample
from uniform population over (O, 0) and let M = Max (X1, X2,
..., Xn). Show that M is a biased but consistent estimator of 0.
Q2. (a) Let X1, X2, ..., Xn be
independently and identically distributed observations from a distribution with
probability density function f0(x). If Y = g(X1, X2,
..., Xn) is an unbiased estimate of 8, then
Use this inequality to show that the sample mean is a uniformly minimum
variance unbiased estimator of mean of a Poisson distribution.
(b) State and prove Chebyshev's inequality. Hence or otherwise, prove that (i)
P[X<l/2]< 4/l and (ii) P[X >2l]
<1/l. if X is any positive random variable having both mean and variance
equal to l.
(c) In a Community of (a + b) potential voters, a are for abortion and b(b<a)
are against it. Suppose that a vote is taken to determine the will of the
majority with regard to legalizing abortion. If n(n <b) random persons of these
(a + b) potential voters do not vote, what is the probability that those against
abortion will win?
Q3. (a) X1 is the time that a customer takes from getting
on line at a service desk in a bank to completion of service and X2
is the time to wait in line before reaching the service desk (X1>X2).
The joint probability density function of (X1, X2) is
given by
Let Y1 = X1 + X2 and Y2 = X1-X2.
Find the joint probability density function of Y1 and Y2.
Are Y1 and Y2, independently distributed? Give reason.
(b) Let X1, X2, X3 and X4 be
independent and identically distributed random variables each having probability
density function
Find the cumulative distribution function and probability density function of
Y = Min (X1, X2, X3, X4).
(c) Using Kolmogorov-Smirnov test, determine whether the sample data given
below come from an exponential distribution with mean 4. The observations in the
sample are
0.7, 5.5, 5.7, 4.3, 0.8, 1.1, 5.1, 4.9, 2.0, 3.9, 5.3, 4.2, 5.2, 5.8, 6.0
(Given, D13, 0.05 = 0.361, D14, 0.05 = 0.349, D15,
0.05 = 0.338)
Q4. (a) Let X1, X2,.., Xn be a random
sample from a distribution with probability density function
f(x) = qxq-1,
0< x <1, q >0
Find the maximum likelihood estimator (MLE) and minimum variance unbiased
estimator (MVUE) of q.
(b) If the random variable X follows normal distribution N(m,
1), then find the Y = 1-0(X)/0(x), where 0 and o denote the cumulative
distribution function and probability density function respectively of N(0, 1),
the standard normal distribution.
(c) Let X1, X2, ..., Xn be a random sample from
a normal population with mean u and variance 25. For a sample size of n=16, find
the uniformly most powerful (UMP) test for testing the simple null hypothesis Ho
: u = 5 against the composite alternative hypotheses (i) H1: u >5 and
(ii) H1: u <5.
SECTION-B
Q5. (a) Y1, Y2, Y3, Y4,
are four independent variables with E (Y1) = E (Y3) =
q1, +q3
+q4, E (Y2) = E (Y4)
= q1-q2
and V(Yi) = q2 (for i = 1, 2,
3, 4). Verify whether q1+q3
and q2,+q3
+ q4 are estimable. If so, obtain their
BLUEs. Also obtain the variances of the BLUES.
(b) Let X = [X1, X2, X3] have trivariate
normal distribution N3 (m,
S), where
Show that (X2-X1)2 + (X3-X2)2
degrees of freedom.
(c)
(d) An investigator selects 10 one-acre plots by simple random sampling and
counts the number of trees (y) on each plot. She also has aerial photographs of
the plantation from which she can estimate the number of trees (x) on each plot
of the entire plantation. Hence, she knows sym mx =
19.7 and since the two counts are approximately proportional through the origin,
she uses a ratio estimate to estimate my. The data
yield the following:
N = 1000 (plantation size), n = 10 (taken by SRS)
yi = The actual count of trees in one-acre plots, i = 1, 2, ..., 10
xi = The aerial estimate for each plot
y = 22.10, x = 20.80
(e) Explain the concept of balanced incomplete block design (BIBD). What are
the conditions for existence of a BIBD? If V = {1, 2, 3, 4, 5, 6, 7), then form
a number of blocks, each of order 3, such that each pair of elements in V is
contained in exactly one block.
Q6. (a) Obtain first canonical correlation and its associated
canonical variable pair for the following correlation matrix :
(b)
(c) Distinguish between a factorial experiment and a number of single-factor
experiments. What is meant by confounding in a factorial experiment? Why is
confounding preferred even at the cost of loss of information on the confounded
effects?
Q7. (a) Explain the method of systematic sampling and give its merits
and demerits. How do you estimate the sampling variance of the estimate of the
population total? Show how systematic sampling is a particular case of cluster
sampling.
(b) Suppose n1 = 11 and n2 = 12 observations are made on
two random vectors X1 and X2 which are assumed to have
bivariate normal distribution with a common covariance matrix
S, but possibly different mean vectors
m1, and m2.
The sample mean vectors and pooled covariance matrix are
Obtain Mahalanobis sample distance D2 and Fisher's linear
discriminant function. Assign the observation X0 = (0, 1) to either
population p1 or p2.
(c) Let y1 = b1-b2
+e1, y2 = b3 +e2
and y3 = b1 +b2
+e3, where e1, e2, and e3 are N
(0,0). Obtain best linear unbiased estimator (BLUE) of (2b1
-b2). Is this linear parametric function
testable?
Q8. (a) Three sides of an equilateral triangle were measured by 5 persons
with the following results :
Is there any significant difference between (i) measurements by the persons
and (ii) the sides of the triangle?
(b)
(c) Derive the expression for the variance of Horvitz-Thompson estimator (HTE)
of population total. Point out the difficulty faced in estimating it. Find
Yates-Grundy variance estimator.
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