l lt'S-2003 I ot a

3
Answer any four par1s:
2 table
a. b
c d
STATISTICS
PAPER·I
SECTION .A
prove that the chi-square test for independence gives
• N<ttd bc2l
-la.+bXa.
whenN=a+ d.
Deline probability generating functir>n
I.
(IOx4
If X and Y arc non-negative integral Independent random variables ·,vith generating
funclions and then prove that the generating function of X+Y is
If x. is distributed according ro F-distribution of dcgrcus of freedom, show U1al
•
P(xH J
Write, in brief, a note on Neyman-Pearson approac.h on of statistical bypotlesis.
Show that the distnl>uuon lor which the charactcnstic funcnon is has the density function

ftxJ
Jt 1-.rz
State the main ;esults you usc for your proof.
State and prove the Borei-Camelllemma
(10x4 • 40)
An urn comains r red and w white balls. k balls are drawn from rhe um and kept aside
without noiicing !heir colour. E'rom r k balls. ooc bail is drawn. What Is
the probability that it is red?
for the Poisson di butiM wiU> parameter pn1ve that
P«t
Un X be a mndom vatiable as'Sw.ui tlg posi uve values u· E (Xl fl. crl and t
pnwethat

•
a
Stale and prove the Rao-Biackwelltheorem
(10•4 • 40)
Le.L .J be a sequence of estimators such that 0 and V 0 as n Sl1ow
tltnt Tis a cons.isteut estimator ol'6
and prove KJ>inchine's theorem for WLLI'f
State Kolmo,gc1tov's SLLN for a sequence of independent random variables.
1 0 1
Let tlte random variable X have the putl.'
j .r) •. r I ami 0 Dcrivu dte SPit test 00 against 61
40)
Let X be an cr1
variable wltere botlt tlte parameters are unknown Determine the LR
test uf Hu. e eo against HI tl fl·l · b3$ed on A $8ntpiB of si?.e n
Describe the Kolmogorov-Smlmov test for goodness of IlL Compare this test with x1-tcst for
goodness of fit..
Star<! ancl prove theLind(berg-Lovy central llntitthcorern,
SECTIONS
Answer uny four pa.rts.
6
7
8.
(IOlC4 40)
In connection with me problem or linear estimation, el'piainthe tenns
estimable parametric function, and
besi linear unbiased estimate (13LVE)
Smte Gauss-MarkofT theorem
If Yt. Y1 <md are observtttions with "commt)n c1 nnd expectations
giYen by












E 01 tl1
..
E e1
prove that m191 • m291 T is estimate if and only iP m1 m1 t
Define. giving examples, the temts 'main effects' and two factors interactions.
Explain the concept of confounding. lllusmne your answer by a suitable example
Establish the rati!;le of a multiple cortell!lion CQCffieienl.
(10" 4
If X N,. and C is a qx p rnatlix of rank prove lbaty C.A N"
Define Hotell ing's T1 -statistic. Show that it is hWarinnt under non-singttlar li near
tmnsfonnalious Met}llou SO!IIC uses of the f
I 4 I 4 10)
Let y be dtstributed as Nr I). Prove t.hat a necessary and sufficient condition for A.Y to
be dfsirlllUled as lis that A1
1.11 a BI.BD v. k and)(. prove that .1 k.. What ltappcns ifv
(10 X 4
?rovet.hat theB!BD witl1 parametel1i v 10, 18. lland i. is not resolvable.
Cons!JJcl a symmetric BlBD with v b k 3 nod i. I
What is missing plor technique Explain i1s importance. In the randomised block desil!n one
observation is missing, Estimate rhe missing value and indicate how you will carry out the
e1<act anyalysis,
Define the J lorvitz.-1'hompsou estimatot and obtain its variance.
(lOx 4 40)
two-stage Sl
npling. Prove Ulat.. if then fLrst-srage.units and them second-stage units
frotu each chosen Jirst-s tage unit selected by simple random sa.tl:lpling. sample mean y is au wtbiased estimate of tlte population mean y Obtain e variance of y .
3 of6
Lxplnin Ute ralio mclhod of eslintalion. Show thai the •arinncc )or the: ro tjo
t!l>.t unalor . sma Jl .:r tha n th at ot. menn lint.I ""tm. wtor p -lC·
2C,
where the notations have lboir usual meaning.
Write a note on double sampling procedure&.

J .
lf I
STATISTICS
PAPER·ll
SECTION A
(Industrial Statistics and Optimisation Tl>chnlques)
1ltempl any four
Explain where and how we use X and R charts for quality control
Ois'""SS the concept of acteptruwe sampling giving sui Ulble e.xamples
0 1
I

101
Discuss ilie .assumptions underlying linear Can be met in real li fe.
situations?
tO)
Explain the concept nf replacemeo1 for failing items Also state various replace men' models.
Explain Hte Markov chain as a method f6rec;fsting
Explain aC<Jcptancesampling for glving assurance about standard deviation of a process
Wltat are cand p cltans Explain the1r relative merits giving e.xamples
What are the problems of life tesring? Discuss one ml)del for lhis problem
Solve followiJJg linear programmiug problem by taking its dual
1fin Z 2x1 T 2xt
Subje<:t to
1
1
l
..t1 X2

10)



14)
A salesman travels ween five ci ties. 11te followmg are. ligures of cost of travel fonn city 1
t6 city
C11
20, C111 4.
cu • s. c ..• 0 6t
c .. c., 20,
further, Cu CJ• and tJte. salesman does not travel between cities where is not given
Solve lbe P.roblem of the salesman lO minimize Iota!. cosL
14)
Explain the concept of game theory. Also discuss a two-person zero-sum game witlt two
allernaffve strategies.
12)
(al
5 ot
J machine costs Rs. 6,000. The run11Tng cost .and resale values of the machine in various
are given Mlow.
Year J 2 3 4
Runing' cost
(in Rs.) 1,000 1.200 1,800
Rcs3le vnluo
(in Rs.) 3,000 1,500 750 375
5 6 7
2,300 2,800 3,400
200 200 200
At wlmt age the machine should be replaced?
14)
At a pubUc telephone booth lhc customer arrival lbllows u Poisson process with average
fnter-anival time 12 'll1e leng1h of the phone call is C'-"Ponentially distributed wi th
an average of 4 minutes. Find the proba bi lity that-
n ani,·al will not have to wait;
a fresh arrival will have to wait for more than I 0 minutes
12)
What Inventory Control'' Discuss various terms used in inventory models.
SECTION&
(Quantitntive Economics aod Official Statistics)
5 Attempt any four·
7
ExplaiP method of forecasting
10)
Brit'fly explain rJ>e Hmitati011< of present system oollecting Ofticial Statistics in I n<Jia,

Discuss the asswnptlous underlyttlg ordinary least squares method uf ostim11Liou and lheir
i111plicaaioos.
10)
E.xplain various gmwlh ruudels used for the srudy ofpopul3tion growth.
10)
Discuss the teeboiqueof path analysis

lb)



Discuss various method$/fonnuJae ror the construction of Index Numbers along with 1heir
relative advantages.

Dfscuss the shapes. propenics and mcthods of litrlog a trend to a time series data in the Jorm
ofY1 K.At•_
What is the problem of heteroscedasticity? Explain itS consect11ences and remedial measures

State >tnd prove lheorem for lin= csti mation
16)
Discuss Ute problem and conditions of ldl!lttifioation in a tancous equations system,

Explain two-stage least squares methpd of estimation
6 ur6

8. E.X]Jiain the of censm 3nd eritiCIIUy <Mtlua te llul of Popul31ion Statistics
thus generated.

no11 •re lite conlru<.1od What p:trnmcten< can studio<l lhtough lhet11 •nd 13ble.
•w usel•1 in the study of demography?
14)
Discuss tho method and uses of Pocior Discuss tb• mc tl•nd and usos of F.1ctor