STATISTICS
Paper—I

SECTION—A
Q. Let the probability that a family has exactly n children be a pn 0 p I). Assume
that a child can be a boy or a girl with equal probability.
Obtain the probability that a family has ^exactly k boys.
Given that a family has at least one boy, what is the probability that there are two
or more children in that family 10
Q. Define convergence in distribution and in mean. Let be a sequence of independent
random variables such that
Xn n3 with probability 1/n2
0 with probability 1-1/n2
Show that Xn 0 in distribution. 10
Q. Define a sufficient statistic.
If X1 and X2 are Bernoulli random variables, examine the sufficiency of T1 and T2
for p where T1 X1 X2 and T2 X1 5X2. 10
Q. x1,x2...xm and y1,y2,...yn be independent random samples from sigma12) and
sigma22) respectively. Find a confidence Interval for sigma12/sigma22at confidence level 1 alpha. 10
Q. Consider n observations X1. X2....... ..., Xn from 0). Obtain the Bayes estimator for
0 under quadratic loss function when a conjugate prior is assumed for 0. 10
Q. How can you find out the discontinuity points of distribution of a random variable if the
characteristic function is given to you Hence find the c.d.f. of a random variable
X whose characteristic function is 1/4 2e2it). 20
Q. Let Xn be a Poisson n 0 O Use the Central Limit Theorem to find the
smallest n such that
<img src='./qimages/1004-2-b.jpg'>
Q. The distribution under the null hypothesis of X is uniform over 2 .... 10 and v r J
the alternative hypothesis, the distribution is given by P{X for k
P{X for k 9 and P{X Obtain most powerful test of size
0.15 and find its power, based on a sample of size one.
Q. A trial can result,in one of three possible outcomes with probabilities p1? p2 and p3
p2 p3 respectively. Construct the Likelihood Ratio test of
against the alternative that these probabilities are different from 1/3 . The test needs to be
constructed on the basis of n-trials. 20
Q. How does one obtain moment estimators If X has the pdf
<img src='./qimages/1004-3-b-1.jpg'>
and a sample of size 50 yields
<img src='./qimages/1004-3-b-2.jpg'>
find moment estimators of 0 and beta. 1 5
Q. Find a Most Powerful test for testing
<img src='./qimages/1004-3-c-1.jpg'>
versus
<img src='./qimages/1004-3-c-2.jpg'>
on the basis of a random sample of size n. Here is the pdf of the random variable
x . 15
q.4(a)
letf(.) be the cumulative distribution function(cdf) given by
<img src='./qimages/1004-4-a.jpg'>
Find the set of discontinuity points of F and express F as a mixture of its discrete and
continuous parts. 20
Q. 4(b)Suppose that X and Y have cdf F and G respectively. Given independent random samples
x2, ..... xm) and y v y2, ..... yn) from these distributions, construct ftfann-Whitney test
of H0 for all t against the aitemative that for at least one t. Also
indicate the test when the alternativei ts f(t)not equaltoG(t) for some t. State the test you would
use when m and n are large. 15
Q. Consider observations from Poisson distribution with parameter X. Develop a Sequential
ProbabilityfetioTest (SPRT) to test H0 :-lamda=lamda0 against H1 :lamda klamda0, k Obtain OC
and ASN functions. . 1 5
SECTION-B
A random sample of size 4 ftom; a biyaiiate normal population provide^ the following
statistics
<img src='./qimages/1004-5-a-1.jpg'>
where X is the sample mean and S is an unbiased estimator of the population dispersion
test the hypothesis HQ where is the population mean vector.
<img src='./qimages/1004-5-a-2.jpg'>
Q. Let X X2 .....Xp)' be a p-dimensional random vector with V(X)=sigma.
Define partial correlation coefficient p 12 between Xj, X2. Show that
<img src='./qimages/1004-5-b.jpg'>
where a is the j)th element of S-1. 1 0
Q. Consider three independent random variables Y2 and Y3 having common variance a 2
and E(Y1)=beta1 beta3, beta1 beta2, beta1+ beta3. Determine the condition of
estimability of the linear parametric function Tp. Obtain a solution of the normal equation
and the S.S. due to error. 1 0
Q. where Vran and Vprop are respectively the variances of the estimated means under simple
random sampling and stratified random sampling with proportional allocation. All other
notations have their usual interpretations. 1 0
Q. 5(e)Describe the layout of a 33 experiment in 4 replicates (with 3 blocks per replicate) using
complete confounding. 10
Q. For an arbitrary fixed effective size sampling design with positive second order inclusion
probabilities, derive the Yates-Grundy form of the variance of the Horvitz-Thompson
estimator of a finite population total and hence obtain the Yates-Grundy unbiased estimator
of this variance. 20
Q. let have the joint moment generating function(mgf)
<img src='./qimages/1004-6-b.jpg'>
Obtain the covariance matrix and the mean vector of X.
Find a constant C such that P(2Xj 3X2 X3 0.95.
Derive the conditional distribution of X1 given X2 x2 and X3 “ x3. ..
(If necessary, you can use P(t 1.645) 0.05 and P(x 1.96) 0.01 where x is a
standard1 normal variate) 15
C-AVZ-0-T00A 8
Q. For the Gauss Markov Model Xbeta, sigma21) the estimator Y is the best linear unbiased
estimator (BLUE) for iff is uncorrelated with all unbiased estimators of
zero. 15
Q. Define a balanced incomplete block design (BIBD). Carry out its intrablock analysis.20
q.7(b)Suppose a random vector X has the covariance matrix
sigma
3 1 1
1 3 1
1 1 5
Find the principal
components of X and obtain the proportion of the total variance accounted for by the
first two principal components. 1 5
Q. Consider the simple linear regression model yi betao beta1Xi e1= i 2,....... n. Show
that if the XiS, are equally spaced (i.e. u iv for fixed values of u and then
Yi Y0 Y1i ci is an equivalent reparametrization (in the sense that both the design
matrices have the same column space). 15
Q. Consider a random sample X1,x2, ....xn from a p-dimensional normal population with ii •
mean vector mew- and dispersion matrix sigma. The purpose of the problem is to construct the
likelihood ratio test for testing MQ I a 2Ip. Find the maximum likelihood
estimate of mew and sigma2. Write down the expression for -21oge in terms of x
and the elements of the sample covariance matrix S. 20
Q. What is meant by confounding in a factorial experiment Why is confounding used even
at the cost of loss of information on the confounded effects Explain the terms ^complete
Confounding’ and ‘partial confounding’. 15
Q. A simple random sample of n clusters, each containing M elements, is drawn from the
N clusters; of the population and the clusters sampled are enumerated completely. Suggest
an unbiased estimator of the population mean per element and derive the expression for
the variance of the proposed estimator in terms of the population i intraclass correlation
coefficient. 15
C-AYZ-O-T00A io