STATISTICS



Time Allowed: 3 Hours Maximum Marks: 300
Instructions
CamlillJItes s1wuJd aJte""tqueslions 1 and5 whkh are compulsory, and Il1!V three o/the remaining
questionssekctingaJkastonefrom eachSection.
AsSllln1 snitnhle tlJl1ll ifconsitkretlnecesswy andintlkaJe /hesnmeckarly. Allqll£stionscarry eqlll11marks.

SECTION A
Answer any four parts among parts to
Choose the correct answer and and answer
A, B and C are three arbitraJY events. The expressiOn for the event that "only A occurs" 1n tenns of Cis
Bn (An Bn (Au Bv
A com is tossed three times in succeSSiOn If A is an event that there are alleast two heads and B is the event that 1st throw gIVes head, the probability P n is
Slale Neyman-Pearson Lemma for testing a Simple hypotheSis agamst a Simple alternative
An estimate Tn based on a sample of Size n is called an asymptotically unbiased estimate of e 1£
limP(T, lim



MalchX with Y y
X

Random expenment Charactenstic function


Sufficlent Non-parametricmethod

InversiOn theorem Prob ability theol)'


Kolmogorovtest Estimate
for go odness 0f fit



Supply the nussmg word(s)

If A and B are lndependent events then the conditional probability is equal


If P q then the binonual distribution is


If a hypotheSiS speclfies the value of all the parameters of a density function, itlS called
Methods which are not concerned with testing or estimating the parameters of a density function of a glven type are called

Obtam the prob ability generating function of the POl sson distribution


Write the prob ability mass function of the negative bmomlal distribution Define a Umfonnly Most Powerful Test (UMPn


Write the expresS! on for ASN function m sequential probability rali 0 test (SPRT)




Obtam the mean and the vanance of the exp onenti al distributi on


Write down the reproductive prop etty 0 f the POisson distribution

Ifx and y have unit vanances and correlation 0.4, find the correlation between x 2y and2x+y


When lS a statistic called the best unbiased estimate of a parameter?




A random expenment conSlsts m taking two balls out of a bag contaimng 3 white and 2 black balls. Write down the corresponding sample space, enumerating all the elemental)' events and th",r prob abilities


8 unblased coms are tossed once. Find the probability that there are at least 7 heads

Obtam estimates of the parameters and of a Nonnal distribution usmg the method of moments


An unbiased com lS tossed 30 times and the followmg sequence of heads and tails en lS observed

HHHTTTHHTTTHHHTTHHHTTTTHHTHHT
Find out the number of runs (rl. Also find the mean E



An unblased com lStossed ti113 heads are obtained. Find the expected value of the number of tosses


State the Weak Law of Large Numbers and write necessal)' and sufficlent conditions for its validity

A bag contains four red and SlX black balls. One balllS taken out a random, without noting its colour. A second b alllS then taken out of the remaimng ball s m the bag at random and found to be black WhatlS the conditional probability that the first ball taken out was also black?


What are the advantages and disadvantages of non-parametric ,nference procedures over corresponding parametric pro cedures?


Show that 2 lS an unbiased estimate of the population vanance,

where for 1 n are observed values m SRSWR sampling and x lS the sample mean


Define Urufonnl y M,rumum Vanance Unblased Estimator 0f a parameter. Expl am the role of Lelunatul-Scheffe Theorem m this context

Write the probability density function of Beta distribution of first kind. Find the value of its r­th moment about ongm


Obtain the Moment Generating Function of the standard Nonnal distribution




SEcnONA
Answer any four parts among parts to
Chao se the most appropnale answer
(il The mam reason for formmg blocks in expenmenla! deSigns is (error control, repl1cation, mcreasmg mductive scope)
The correlation coefficient of y on x is 0.5 This means the proportion of the van ability m p accounted for by x is
In a deSign, the mam effects are confounded with the block effects. (Split Plot, Latin Square Randomized Blocks)
In a sample of Size drawn from a population of Size drawn by the method of Simple random samplmg Wlthoul replacement, the probability that a particular UnitlS
mc1uded is N/n.
MalchX with Y y
X
(il Replication Stratified sampling

Neyman allocation AnalySiS ofvanance
Vanance covanance matrix DeSign of expenmenl



One-way c1asslfication Multivanale analySiS



Fill up the blanks m parts and and answer


The arithmetic mean of apositive valued statistical van able lS than the reaprocal of the arithmeti c mean 0f the reapro cal of the van able


If one observation is mlSsmg m a Randomlzed Block DeS!gn with v treatments and b blocb m the AnalySiS of Vanance Table, the number of degrees of freedom for treatments w111 be and that for error will be

The Gauss-Markov Theorem reqU1res the error components to have mean values, vanance and all palr-WlSe covanance's to be


Wnte down the structure 0 f the AnalySiS 0 f Van ance Table for a Latin Square DeS!gn oforderm




Write down an approXlmation for the vanance of the usual estimate 0 f the ratio 0 f the means of two correlated vanables, m the case ofSRSWR sampling


Write down the vanance of the Horvitz-Thomson unbiased estimate of the population total

Write the linear statisti cal model for the analySiS of Randoffilzed Blo ck DeS! gn


Write down the partial correlation coefficlent between the first and second elimmating the effect of the third vanable from each of them, m tenns of the three palr-WiSe total correlation coefficlents




Define two-stage sampling


Describe the use of critical differences m companson of pair-WiSe differences amongst treatment effects

IfregresSlon coefficlent of yon x lS -0.5 and regressiOn coefficlent ofx on y lS find the value of correlation coefficlent


Explain the concept 0f confounding m factonal exp enments




Calculate the expectation and the vanance 0f the sample mean m SRSWOR samplmg
Construct a Latin Square DeSign 0f order 5
Describe an appropnate test for equality of means of several populations on the baslS of
mdependent samples drawn from them
Interpret the AnalySiS 0f Vanance Table ln two-way nuxed effects model with equal numb er
of observations m each cell
Explam how the baslC pnnclples of randomlzation, replication and error control are utilized
m a Randomlzed Blocks DeSign
WhatlS the difficulty m analyzmg a Latin Square DeSign of Size
Give your critical comments on the followmg observation "A great disadvantage of
systematic sampling lS that the vanance of the estimated mean does not necessarily decrease
as the sample Size lS mcreased
Show that the con-elation matrix lS atl east po sitive senu -definite
Explam Fixed Effect and Mixed Effect mo dels m two -way clasSification with examples
For a BIBD prove that J..(v r
If x 4y 5 and y (III x 4 are the regresSion lines of x on y and y on x respectively,
find the value of the 4 correlation coefficlent between x and y
In standard notations, giVen that
r12 0.5, r13 0.4 and r" 0.2. calculate R123