CS (MAIN) EXAM, 2010 STATISTICS Paper-I
Time Allowed Three Hours Maximum Marks 300
INSTRUCTIOS
Each question is printed both in Hindi and in English. Answers mu.st be written in the medium specified in the Admission Certificate issued to you, which must be stated clearly on the cover of the answer-book in the space provided for the purpose. No marks will be given for the ansiuers written in a medium other than that specified in the Admission Certifi.cate. Candidates should attem.pt Question Nos. i and S UJhich are com.pulsory, and any three of the remaining questions selecting at least one question from each Section. The number of marks carried by each question is indicated at the end of the question. Assum.e suitable data if considered necessary and indicate the same clearly. Notations and syrnbols used are as usual.

Section-A

1. Answer any five parts of the following ·12x5=60
residents of a locality are earning more than Rs 10 lakhs per annum. The percentage of single earner in the family among those earning more than Rs 10 lakhs per annum is 80o/o, whereas percentage of single earner among others (earning less than Rs IO lakhs per annum} is 50%. A randomly selected resident of that locality was found to be single earner. What is the probability that he earns more than Rs 10 lakhs per annum?
T is a triangle in plane with vertices ROOT and (root2, root2). Let denote the area of intersection of T and the set S x,. b y}. Does F define a cumulative distribution function? If yes, find the joint probability density function.
Two players A and B decide to play a series of at 1nost 7 games. A player who wins 4 games wins the series. If both the players have equal chance of winning a game and A wins the first two games, what is the probability that A Will Win the series?
A city has N taxis numbered I to N. A person standing at a crossing noted n un1, bers of n taxis that pass that crossing (same taxi does not pass twice). If Mn is the highest taxi number noted, find an unbiased estimate of N based on M n .
Let X 1 X 2 •... X n be the lifetime of n randomly selected patients of cancer. An estimate is required for the probability that a patient survives beyond time i.e., P(X Find the maximum likelihood estimator of if lifetime X of the patients follow the following probability density function
f 2x/beta exp(-x 2 beta), if x 0
0 otherwise
Let U x be the number of Y,s those are smaller than in independent random samples X1 X 2 ... X n and Y1 Y2 ... Y m. Find E(U x

2. Cumulative distribution function (c.d.f.) of a random variable X is
0 if x 0
if 0
if
1 if
Find a such that we can write where and F d are c.d.f. of continuous and discrete random variables respectively. Find Fc and Fd 20

The joint probability density function of random variables X and Y is given as
k(y2 x 2 e-Y if Ix y y 0
0 otherwise
comment on the indipendence and Correlation between X and y. 20

Explain consistency of estimators. Prove that second smallest observation in a random sample of size n from the following probability density function 1s consistent estimator of B 20
f 1 if X 0
0 otherwise
3 . The joint probability density function of X and Y is
if y
0,elseWhere
Derive the distribution of X Y. 20
Explain convergence in rth mean and convergence almost sure.Check various modes of the convergence for {xn}.Where Xn's are independently distributed as follows
1 1/nr and P[Xn 1/n r
where r and n i ...
State and prove Wald's fundamental identity of sequential analysis. 20

4. The following is a random sample of 10 numbers lying between
0·22 0·90 0·96 0·78 0-66
0·18 0·73 0·58 0·II 0·97
We wish to test the hyp othesis H O sample comes fr om uniform distribu tion on 1). Explain how you win use Kolmogorov- Smirnov test of goodness of fit fo r HO • Obtain the valu e of te st statistic .
A random observation X belongs to a populati on .having probability density fu nction f Des cribe likelihood ra tio test fo r testing H0 f1 against H1 f 2 based on a single observation when <img src='./qimages/907-4b.jpg'>
X 1 X 2 .•• X n is a random sample from Bernoulli population with parameter 8. Consider that 8 has beta prior distribution with hyper-parameters(a,b) . Obtain Bayes estimator for 0 under quadratic loss function and hence obtain min max estimator for 9. 20

Section-B

5. Answer any five parts of the following 12X5=6Q
Consider a population of 3 units U1 .. U2 and U3 • The size measures of U1 U2 and U3 are respectively X1 6., X2 4 and X3 2. A sample of two units is drawn from the population without replacewmiethn t psurocbha tbhilaitty t hpe rfiorpsotr tuionnita il s tsoe letchteeidr sizes and second unit is selected with pthreo braebmilainity ipnTgo punortitiso.n alIf nto the sizes of i denotes the probability of inclusion of Ui in the sample., show that
pie1=51/60,pie2=44/60,pie3=25/60

X X2., has 3-variate normal distribution with mean vector µ and covariance matrix sigma, where <img src='./qimages/907-5b.jpg'> Show that x1 and x2 are not independent but [Xand X 2 is independent of x3.Further show that 1/2 (X1 X2 and X3 are also independent.

X [X1 X2 X3 has 3-variate normal distribution with mean vector µ and dispersion matrix sigma,where <img src='./qimages/907-5c.jpg'>
<img src='./qimages/907-5d.jpg'> Fit the equation Y b0 biX1 b2 X2 and calculate multiple correlation coefficient.
In a population consisting of three units Y1 U2 and U3 the observations are si1z, eY 22 anisd dYr3a wrens pfercotmiv eltyh. eA psoampuplaltei oonf by using simple random sampling without replacement. es ti mator T as follows: <img src='./qimages/907-5e.jpg'> Show that T is unbiased estimator of population mean and has smaller variance than sample mean if Y3 Y2 3Y1 Y3 0.
Explain 1n detail the procedure of randomization (i.e . allocation of treatments ran domly to plots) followed in randomized block design and Latin square design .

6. Let X X 2 ... X P denote a column vector of variables X i . Prove that Y fo llows a p-variate normal distribution N P sigma) iff there exists a random X (X1 J X 2 ... X P of independent standard normal vari ables Xi "s such that Y µ BX with pr obability one fo r som e matrix B of full rank and BB sigma. 20
Distinguish betwe en 2 3 and 3 2 fac torial exp eriments . Explain how you will ge t sum of square s due to main and interactions in 3 2 -factorial experiments . Give format of AN OV A table fo r 3 2 -facto rial expe riment mentioning degrees of fr eedom if ran domized block design having r blocks containin g 9 plots each is used . 2 0
From a population of N units, a random sample of 3 units is drawn by simpl e ran dom sampling with replacement . Find the expectat ion of distinct uni ts in the sru:nple . Define T as mean of distinct uni ts in the sample. Show that T is unbiased estimate of populati on mean. 20

7. Let X be dis tribu ted as Np p-variate normal distribution) . Prove that Y CX where C is r x p matrix of rank r r is distributed as Nr Csigma 20
The vari abl e X under study has rec tangular distri bution in the interval d). The interval is divided into k equal subintervals which fo rm k strata of equal sizes. Fro m each stratum a simp le ran dom sample of n k units is drawn . Let Vj and V2 be variances of estimator of populati on mean based on stratified and unstratified sample$ of size n respective ly . Prove that Vi /V2 k-2 . 20
In a randomized block design with okb sterervaatmtioennst s araen dm irs sinregp.l icOantieo nmsi, sstiwnog jthobs ebrvloactkio ann ids trhelea toetdh etro iisth r etrlaetaetdm teon at tinh Htreowat mwilentl iyno u'3t h ebstloimcka te(i tah ean dm 1js'#-s1f3n)g. tohbes eervstaimtioantes Df emriivses itnhge oexbpserervsasitoionnss f.o r 20

8. Define ratio estimator for estimating the population total and derive expression for the standard error of the estimator. 20
Obtain an expression for estimated value of Y for given X x0 by fitting a curve e x Y a be 2x to n given points (xi Y£ i ... n using least-square method. 20
<img src='./qimages/907-8c.jpg'> prove that T1 and T2 are unbiased estimators of new and

Note English ve rs ion of th e Ins truc tions is printed on the front cover of this question paper .