IAS Mains Statistics 1990

7/28/2019 IAS Mains Statistics 1990

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Time allowed : 3 hours Maximummarks: 300

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SECnONA{Probability)

State and prove the addition theorem of total probability after explaining the te ob ity.A coin is tossed until for the first time the same side of the coin appears ~ ession.To every possible outcome requiring n tosses attribute probabuT 'Vd ind theprobabilities of the following events:(i) the experiment ends before the 6th toss(ii) an even number of osses is required for the exp ~(iii) an odd number of osses is required for the r _ ~Explain a priori and a posteriori probabilities. State : : . ~ B a y e s ' theorem.Let X be a random variable and define ~ ~ w that(i) F is non-decreasing ~(ii) Lt F(x)=l,_,_

Lt F(x)= 0 flj_,_15+30+15

(iii) F is right ~ s.Let X be a ~ whose eli stribution function F(X) is continuous and strictlyincreasing on ~ t Y=F(X). Show that Y has a uniform distribution on . 9 { ~ 1 .Define ch a c t e n ~ c t i o n of a random variable X. The characteristic function of arandom v ble !+known to vanish at a point a. Show that the characteristic function ofX+Zalso v a; where Z is a random variable independent ofX.

20+20+203. (a) d prove Chebysheffe' s inequality. Show that ifx is a random variable such that ifX is

(c)

m variable such that, P(XO,P[ ~ c] E (x)

State, explain and pro:e the Lindeberg-Levy form of he Central Limit Theorem.If (X,. be a sequence of independent random variables with

IP[X =+I]=P[X =-1]=- n=l 22' ...show that

p =I{max Ix1+ :12 + ..., +Xt I}k ~ n} .: . .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:. .:.:;


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24+24+12What is a binomial distribution ? In a binomial distribution (f (k; r, p )}, let ~ O and r ~ insuch a way that rp = 1.. remains fixed. Show that

f (k ; r, p) ~ p(k, 1..)where (p (k, 1..)} is a Poisson distribution,Give the statements of weak and strong Iaw of large numbers. Discuss the significance ofthese laws.Let (X,. be a sequence of mdepen dent standard normal vanate, Prove ~n-I (XI+x2 + +:x,)converges almost surely to zero as n ~ 0c:::;; 20+20+20

SECnONB +{Satistical Inforence) 0

Explain the concepts of(i) UMPU tests;(ii) Simi! ar regions;(iii) Type A1critical region.Show that the MP and UM P tests o

'7>c:::;;~ a s e dState Neyman and Pearson's furl en lemma and apply it to obtain the test for testingcJl=i aganst OJ> I, whenr(1p IS from N[O, cJl] Is th1s test UMP? Is 1t unbiased?G1ve reasons

15+15+30Descnbe the ~ m ch non-parametric tests are appropnate D1scuss the commonnon -parametnc c onFmd the ean an, ance of the run test statistic under the null hypothesis Discuss itseffici en \ h ~ to Student's t-test.

~ " " 1 1 1 1 . - R pnnc1ple of constructing tests Show that these tests arc consistent underond1t1ons State these condi tlons clearly18+ 18+24

(c) Given a random sam pie of n from a Bernoulli an population with constant probability p. If weassume that the sample estimate of p, X, is approximately N [ p p ( ~ p)J how that theshortest confidence interval for p is [p 1


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24+16+20What is a sequential test ? How will you develop an optimum test of a specified strength for asimple null hypothesis versus a simple a! ternative ?Find expressions for the sample size expected for termination of SPRT both under Ho and H1.Clearly state all the assumptions made.A random variable follows the normal distribution N[8, d], where d is known. Derive theSPRT for testing Ho : ~ E b against H1 : ~ 8 Obtain the approximate expression for the OCfunction.

SECTION Ill{Linear Info renee and Multivariate Analysis)

Discuss in details the 'method of!east squares' and its usefulness.Let there be hk observations, each ofwhich can be specifi ~ the categories of twofactors, A with h and B with k categories. Let the obj dy be the differences incategories of each factor. Set up the suitable linear ~ iving all the details obtainsui table analysis of variance table. Specify the hyp ese d explain underlying teats ofsignificance.

"' 'ooll'"'u the estimation the regression coefficients andd i : l : ~ ~ ~ ~ , ~ t l l l e s s 1 i o n coefficient. Outline the relevant tests ofn1 and the estimated regression equation.

H l e n n c ; a ~ normal variables. If

Derive Fisher's discriminant function and45 + 15

For the following trivariate normal distribution having densityc exp [-Y. ( i + 4y2 + 9r- 2xy + 4yz)]


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where c is constant, obtain (i) the multiple correlation co efficient of X on Y and Z; (ii) thepartial correlation coefficient ofX and Y when Z is held fixed.

(b) Obtain the null distribution of the sample correlation coefficient r, based on n observationsfrom the bivariate normal distribution. Further, show that the statistic r.Jn- 21 .. 1- r2follows Student t-distribution on (n-2) degrees of freedom.

(c) Obtain a confidence interval for a linear combination of k independent estimable parametersin the usual linear model. ~uO

,Ja&'(j0 ~+

~


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(a){Engineering Satistics)

Explain the statistical basis and construction of (i) control charts for mean and range, and (ii)p and np charts. How is the choice between p and np charts made?(b) An item is made in lots of 200 units each. The I ots are given I 0 0% inspection. The recordsheets for the first 2 5 lots inspected showed that a total of75 units were defective. Determine

the tria! control limits for a np-chart.(a)

(b)

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Describe single sampling plan for attributes. Obtain OC and ASN functions for this plan andexplain the terms:(i) Producer's and consumer's risks.(ii) AQL and LTPD.Lots which are 0.5% detective are submitted for acceptance.found to be satisfactory by the consumer. Which plan would you recommso that it may involve least cost, when the probability of acceptaneach plan?

producer,is given for

::: :::::: ::;: ::; ::::; r fP Define (i) reliability function, (ii) hazard u n c t : l ~ denst ty funct:lon and (tv) meantime to failure (MTTF). Obtain relations amonFor hazard functions (i) h(t) = IJ., IJ.>O h(t 1..>0, compute the reliability functions::e:: ut on a life lest ~ nt and the test is terminated after the failure ofr items (I 5, r 5, n).If the failure time x of an ~ the exponential distribution with mean a, obtain themaximum l i k e l i h o o d ~ a!Ou.Define a series systet a!lel system. A system consists of four identical componentsconnected in p arB must be the reliability ofeach component, if the overall reliabilityof the system is .9If a syst of fiv ponents is connected in series, find the mean time between failures ofthe sys ~ components have failure time exponential eli stributions with failure ratesAt, J..2, 5 respectively.

ndancy? Explam the use of redundancy m rehabthty Improvement

SECTION Ill

(b) Solve the following linear programming problem by simplex method:Maximize

z = -2x1 - x2subjected to

3x1+ x2 = 3


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SEC110NV{Demo?J'aphyand Psychometry)

Explain crude and standardised death rates. In what way is standardised death rate superior tocrude death rate? Describe briefly the direct and indirect methods of finding standardiseddeath rates.

(b) Find standardised death rate by direct and indirect methods for the data given below:

(a)

(b)

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Standard PopulationAge Population Specificin ('000) death rate0-5 8 50

5-15 10 1515-50 27 10

Population APopulationin ('000)

12

950 and above 5 60 59

Age ittyears

45

lx Lx

d explain how a complete life table__ ain the rei ation ship between different

Tx eo

4,850,300

omp ar de birth rate, gross and net production rates as measures of fertility.:Pst Jaw as a mathematical representation of population growth. Describe ai'ilg a logistic curve.

le population theory.l l . ' - ~ " ' l am briefly the concept of reliability of scores in educationa! and psychological

What do you understand by a T-score? Explain clearly the method of converting raw testscores into T-scores.(c) Describe the Kuder-Richardson method ofassessing the reliability of a test.(d) What do you mean by intelligence quotiem (IQ)? Describe the procedure and tests formeasuring IQ.

IAS Mains Statistics 1990

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