If F(x) is the distribution function of the random variable with ュ and variance cr 2, show

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Marjorie Fleming
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2 4. sinnt 9\,= nt Determine the distribution function of X. and hence show that even though the sequence of characteristic function 4\, (t) converge to a limit 4> (t), the sequence of distribution functions does not converge to a distribution function. What is the condition that is violated here? If F(x) is the distribution function of the random variable with ュ and variance cr 2, show that N < forx<ji セG@ F(x-JI) ikhxセjii@ c;o > forx>ji I セHKi@x- Jl that Z = セ Q@ /;.2 - セ S@ I;. has a s power function, critical region and level of significance. Give an account of セ BQBG G B ケ@ -Pearson theory of testing a statistical hypothesis and show how you would a most powerful test for testing a simple null hypothesis against a simple alternation. Show 7. Discuss the likelihood ratio criterion procedure of test construction with the aid of an example. Distinguish a non-parametric test from the usual test procedure. Derive the run test, rating clearly the hypothesis tested and the assumptions made. Discuss the large sample form of the test.

3 8. Show that the sequential probability ratio test of a simple hypothesis against a simp! e alternation terminates with probability one. 1/3 It is desired to test the hypothesis Ho : a= 113 against the alternative H1 : a= 2/3, on a Bernoulli random variable X with P[X =I] =a= 1-P[X=O], sequentially. SECnONB 9. State and prove Gauss-Markov theorem equations" are always consistent. stimation and prove that "normal 10. Explain the notion of partial co ficient in relation to n random variables and If all the total correlatio' c: ents pij are equal to p show that every partial correlation coefficient with k ウセウ is pi! + kp. Show also that in this case, the multi pie correlation co ヲヲ W セ S L@....) of any variable with the other (n-1) vari abies is given by I H pi{z セ ] セ [}@ 11. H Hi elling's r =statistic based on N independent observations on a P-variate normal N>P. Discuss the uses of this statistic and its asymptotic distribution. Given a sam pie of N observations from a P-variate normal distribution, what are the ML ウセ of the mean IJ. and covariance matrix L: of the distribution? Show that the sample mean X is distributed normally and independently of the ML estimate of L:. 12. Explain "Discriminant analysis". Derive the best liner function to discriminate between two multi variate normal populations with the same dispersion matrix. Explain how Fisher's discriminate function is related to Mahalano bi s-d 2 and Hotelling' s-r.

5 8. Explain (i) (ii) (iii) Hazard function, Reliability function and Redundancy. The life of an item follows one-parameter exponential clistri bution and it items are put on test. The experiment is terminated when r of them has failed, and let X1 5, X 2 5,... 5, X, denote their failure times. Derive an estimate of the reliability function. SECTION Ill Explam Simplex procedure for sohnng a linear programnung e pro bier Usmg Charnes method, numm1ze z = 2x1+x 2, subject to the constra1nv 3Xi + Xl = 3, 4Xi + 3X2;:: 6, Xi+ 2X2 5, 3, Xl;:: 0. What is transportation problem? Explain Vogel's approximation dt ennine an initial basic feasible solution of a transportation problem. Solve the folio ation problem: Origin Capacity &quirement For an M/Mil セ ine in steady state, (i) the probability that at any time, there are at units in the s. 1i) expected number of units in the system. An item i roduce the rate of 50 items per day. The demand occurs at the rate of 25 items per dav f e sef:;p cost is Rs. I 00 per production run, and the holding cost is Re. 0.0 I per _,_..nd the economic lot size for one run, assuming that the shortages are not determine two -step transition probabilities. Explain the terms: (i) FORTRAN constants,

(ii) (iii) FORTRAN variable names, arithmetic expressions. Explain the execution of DO statements. What rules are to be observed for using DO statements in a FORTRAN programme? 13. 14. 15. 16.

6 (ii) (iii) FORTRAN variable names, arithmetic expressions. Explain the execution of DO statements. What rules are to be observed for using DO statements in a FORTRAN programme? SECTION IV (Quantitative Economics) Describe ratio to trend and link relative methods for measuring seasonal fluctuati series and discuss their merits and demerits. Describe variate difference method for estimating the variance of the イ ent in a lime senes.,'-j. Define Laspeyres', Paasche's and Marshall-Edgeworth index イエZ and show that Marshall-Edgeworth index number lies between. Lasp eyres' and p セク@ numbers. ゥョウKクセKL ] ケ@ Describe the uses and limitations of a cost of living. dicate important steps at'8 involved in its construction. What is a demand function? Describe normal condi@ s o and. If AR and MR denote the average revenue and marginal revenues & ut, en show that the elasticity of demand for the pro duct is given by ARf(AR-MR). In a two-variate linear model, the obse abies X andy are given by X= x+u, Y=y+v where x and y are true values of r pectively u and v are observational errors. Describe a method of obtaini co ten estimators of the parameters in the model where sis the stfsst istur ance. For the following income d e model, obtain consistent estimators of the parameters using two -stage least ウアオセウ@ e : Ct=a.+ セ ylk Yt=Ct+Zt. t= 1. 2,... n '(../ en and obtain rank and order conditions for the identifiability of a sing! e Ii simultaneous equations system. SECTIONV Describe the direct and the indirect methods of 18. Explain various columns of a life tab! e and relations between them. What do you mean by fertility of a population? Define crude birth rate and general fertility rate. Discuss their relative merits and demerits. Define gross reproduction rate (GRR) and net reproduction raft (NRR) and explain in what way they differ from one another as measures of reproduction. How does NRR indicate the growth of the population?

7 19. Distinguish between Z-scores and T -scores. Explain clearly the method of converting raw test scores into T-scores. 20. Write notes on: (c) Discuss various methods used to determine the reliability oftest scores. Population growth curves. Lotka' s stable population theory. IQ tests. +

I Time Allowed : Three Hours I I Maximum Marks

セ G@.. M セエ @ STATISTICS D-GT-M-TTB Paper-11 I Time Allowed : Three Hours I I Maximum Marks 200 I INSTRUCTIONS Candidates should attempt Question Nos. 1 and 5 which are compulsory, and any THREE of the