Define characteristic function. State its properties. State and prove inversion theorem.

Transcription

1 ASSIGNMENT - 1, MAY 013. Paper I PROBABILITY AND DISTRIBUTION THEORY (DMSTT 01) 1. (a) Give the Kolmogorov definition of probability. State and prove Borel cantelli lemma. Define : (i) distribution function (ii) p.m.f. and (iii) p.d.f. Let ( X, Y ) be a random variable of continuous type with p.d.f. f. Let Z = XY and if p( y= 0 ) = 0, let V = X / Y. Obtain the p.d.f. is of Z and W.. (a) Let X and Y be jointly distributed with p.d.f. f 1 + xy = x < 1, y< 1 4 = 0, otherwise ( x, y), Obtain the marginal densities of X and Y. Show that X and Y are independent. Define characteristic function. State its properties. State and prove inversion theorem. 3. (a) Define : (i) convergence in law (ii) convergence in probability and (iii) almost sure convergence. Establish their interrelationships. State and prove Khintchine s WLLN. Examine whether WLLN holds for the following 1 λ sequence of independent random variables : p( xn = n) =. n = p( xn = n), λ ( x = ) = n p n (a) State and prove Levy and Lindberg form of central limit theorem. State and prove the string law of large numbers for a sequence of Bernoullian trials. 5. (a) Define a multinomial distribution. Obtain its m.g.f. Hence obtain ( x, ) cov. What do you mean by a compound distribution? Obtain the probability function of compound poisson distribution and identify it. i x j

2 ASSIGNMENT -, MAY 013. Paper I PROBABILITY AND DISTRIBUTION THEORY (DMSTT 01) 1. (a) Define Laplace distribution. Obtain its characteristic function. Hence obtain its mean and variance. Define Weibull distribution. Obatin its characteristic function. Hence obtain its mean and variance.. (a) Define the log-normal distribution. State and prove its reproductive property. Define logistic distribution obtain its m.g.f. 3. (a) Derive the distribution of non-central chi-square. Derive the joint distribution of the th j and th k order statistic for 1 j < k n. 4. (a) Derive the distribution of non-central F and hence show that central F is a particular case of it. Derive the distribution of central t. 5. Write short notes on any Two of the following : (a) (c) Chebyshev s inequality. Kolomogorov s strong law of large numbers. Compound binomial distribution. (d) Interrelationships between chi-square, t and F. (e) Cauchy Criterion.

3 (DMSTT 0) ASSIGNMENT - 1, MAY 013. Paper II STATISTICAL INFERENCE 1. (a) Define MVUE and explain a method of getting MVUE. Explain the concepts of sufficiency and minimal sufficiency. Describe how minimal sufficiency is related to bounded completeness.. (a) State and prove Cehmann-Schffe theorem. Obtain the general form of distribution admitting sufficient statistic. 3. (a) Explain the concepts of CAN and CAUN estimators. Explain the construction of CAN estimators based on moments. Describe the maximum likelihood method of estimation. Find an ML estimator for f x, θ = 1 + θ x, 0 < x < 1, θ > 0 based on a sample of size n. the parameter θ in ( ) ( ) θ 4. (a) Explain the general method of obtaining confidence limits. Describe the criterion for the selection of a confidence interval from among infinite set of confidence intervals. Define : (i) (ii) Efficiency and Consistency. Show that under certain regularity conditions to be stated by you, Cramer- Rao down bound serves as a lower bound to the CAUN estimators. 5. (a) Distinguish between non- randomised and randomised construct MP critical region N M,σ with M known but σ is unknown. for a random sample from ( ) Derive the asymptotic distribution of the LR test criterion.

4 (DMSTT 0) ASSIGNMENT -, MAY 013. Paper II STATISTICAL INFERENCE 1. (a) State the prove Lehmann-Pearson Lemma. Give its importance in testing of hypothesis. Construct a LR test to test H 0 : µ = µ 0 against 1 : µ µ 0 where both µ and σ are unknown.. (a) Describe Kolomogolor-Sminnor one sample and two sample tests. Explain Mann-Whitney U-test. 3. (a) Show that SPRT terminates eventually with certainty. H in sampling from N ( µ,σ ) Define OC and ASN functions of SPRT. Derive them for testing the proportion of a binomial distribution. 4. (a) State and prove Wald s fundamental identify. Describe Wald s SPRT. Derive OC and ASN functions for testing the mean of a normal distribution with unit variance. 5. Write short notes on any TWO of the following : (a) (c) (d) (e) Factorization theorem. CAN estimators. Monotone likelihood ratio and UMP tests. Median test. SPRT for testing Poisson parameter.

5 (DMSTT 03) ASSIGNMENT - 1, MAY 013. Paper III SAMPLING THEORY 1. (a) What is a sample survey? In what respect is it superior to a census survey? Discuss the basic principles of a sample survey. Describe the functions of NSSO.. (a) Under what circumstances can complete enumeration be recommended in preference to a sample survey? Discuss the main steps involved in a sample survey. What are the different sources of errors in a sample survey? Explain. 3. (a) Define random sampling. Why it is preferred to other methods of selection of sample? How will you estimate the mean and its sampling error from a random sample from a finite population? In the usual notation show that V ( yn ) R V ( yst ) P V ( yst ) N. 4. (a) With usual notation find the variance of Pst, the estimate appropriate to stratified random sampling for the proportion in the whole population. Hence determine the best choice of the nn in order to minimise r(pst). Derive the formula for n with continuous data. Obtain the estimates the sample sizes in the respective strata under optimum allocation. 5. (a) Define cluster sampling. Determine the optimum cluster size so as to minimize the variance for a fixed cost. Explain the systematic sampling discuss its advantages and disadvantages. Prove that the systematic sample mean is more precise than the srswor under the condition to be stated by you.

6 ASSIGNMENT -, MAY 013. Paper III SAMPLING THEORY (DMSTT 03) 1. (a) Explain a procedure of drawing a sample by two stage sampling and give an example. Describe the applications of two-stage sampling. In sampling with PPS with replacement obtain the estimator of the population total. Derive its variance.. (a) What is PPS sampling? Explain Cahiri's method of PPS sampling. In PPS sampling with replacement define an unbiased estimator of population mean. Derive the variance of the estimated mean in two-stage sampling. 3. (a) Describe ratio estimation in stratified sampling. Define a combined ratio estimator for the population total and obtain the expression for the large sample variance of the estimator. Explain simple regression estimate of the population mean. It is unbiased? If not point out when it is unbiased? 4. (a) Discuss critically the use of auxiliary information with theoretical justification supported by some illustrate examples. Discuss the relative efficiencies of ratio and regression estimates. 5. Write short notes on any two of the following: (a) (c) (d) (e) CSO. Gain in precision due to stratification. Multistage sampling. Random number tables and their use. Bias and mean square error of ratio estimator.

7 (DMSTT 04) ASSIGNMENT - 1, MAY 013. Paper IV DESIGN OF EXPERIMENT 1. (a) Define the following terms with (i) (ii) Determinant of Matrix Idem potent Matrix (iii) Characteristic roots of a Matrix (iv) Derivative of a Matrix. Explain the rank under conditions of a linear model in deciding its consistency.. (a) State and prove Cuctroms theorems. Explain the role of Cornley Hamilton theorem. 3. (a) State and prove Gauss Markov theorem. What is the estimability of a parametric function? Explain with an example. 4. (a) Show that in a linear model ~ ~ ~ Y X + E 1 1 =β the OLS estimator of β is = ( X X ) X Y ^ β is BLUE. Stating the assumptions clearly explain general linear model. 5. (a) Explain the testing procedure associated with equality of K means. Discuss the F-test associated with analysis covariance with one way classified data.

8 (DMSTT 04) ASSIGNMENT -, MAY 013. Paper IV DESIGN OF EXPERIMENT 1. (a) Distinguish between random and mined effect models. Explain the linear model associated with analysis of Co-variance with two way classification.. (a) Discuss the principles of design of experiments. Derive the statistic associated with testing the equality of k treatment effects in CRD. 3. (a) Explain the layout associated with Graccotchion square design with an example. Explain the procedure associated with utlising mutually of the regional latin squares in designing an experiment. 4. (a) Discuss the role of factorial experiments in industrial statistics. Derive the estimates of marins and interaction effects in 3 factorial experiment. 5. (a) What is balanced incomplete block design? How do you analyse it? Explain in detail the uses of 3 3 factorial experiment.