(DMSTT 01) M.Sc. DEGREE EXAMINATION, DECEMBER First Year Statistics Paper I PROBABILITY AND DISTRIBUTION THEORY. Answer any FIVE questions.

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1 (DMSTT 01) M.Sc. DEGREE EXAMINATION, DECEMBER First Year Statistics Paper I PROBABILITY AND DISTRIBUTION THEORY Time : Three hours Maximum : 100 marks Answer any FIVE questions. All questions carry equal marks. 1. (a) Give the axiomatic definition of probability. State and prove Borel-Contelli lemma. Define a random variable. Prove that the function f x p w : x w x for all x R is a distribution function. 2. (a) State and prove Holder's in equality. Hence s r 1 r s prove that E x 1 E x, 1 < r < s <. Let (x, y) be jointly distributed with p.d.f. f(x, y) = 2, 0 < x < y < 1, and = 0 otherwise obtain the (i) marginal densities of x and y (ii) E (y/x) and V(x/y).

2 3. (a) Define (i) convergence in law (ii) convergence in probability and (iii) almost sure convergence. Establish their interrelationships. State and prove Lindberg-Levy form of central limit theorem. 4. (a) State and prove Kolmogorov SLLN. State and prove De Moivre's form of central limit theorem. 5. (a) What is the compound binomial distribution? Obtain its probability function and identify it. Define a multinomial distribution. State and prove the reproductive property of multinomial distribution. 6. (a) Define Weibull distribution. Obtain its m.g.f. Hence obtain its mean and variance. Define the log-normal distribution. State and prove its reproductive property. 7. (a) Define Laplace distribution. Obtain its characteristics function. Show that the mean demination about mean is 1. Define logistic distribution. Obtain its m.g.f. 2 (DMSTT 01)

3 8. (a) Define order statistics. Derive the joint distribution of order statistics for x 1, x2, x3... x n. Hence obtain the marginal p.d.f. of r th order statistics. Derive the p.d.f. of non-central t-distribution and show that central-t distribution is a particular case of it. 9. (a) Derive the non-central Chi-square distribution. Derive the distribution of central F. 10. Write short nsotes on any TWO of the following : (a) Chebyshev's in equality Liapounov's form of CLT (c) Compound Poisson distribution (d) M.G.F. of log-normal distribution (e) Order statistics. 3 (DMSTT 01)

4 (DMSTT 02) M.Sc. DEGREE EXAMINATION, DECEMBER First Year Statistics Paper II STATISTICAL INFERENCE Time : Three hours Maximum : 100 marks Answer any FIVE questions. All questions carry equal marks. 1. (a) Explain sufficiency and complete sufficiency. Show that for a random sample from b ( 1, ) the complete sufficient statistic for is n X i i 1. State and prove Lehmann - Scheffe theorem. 2. (a) Define UMVU estimator and state Rao Blackwell theorem. Explain how Rao -Black well theorem can be used to obtain the UMVU estimator. State and prove Fisher-Neymann factorisation theorem.

5 3. (a) Define m. l. estimator and moment estimator. let x 1, x2... xn be i i d random variable with the p.d.f f( x, ) given by if x [, 2 2 f ( x, ) obtain an 0 otherwise MLE for. Is four MLE consistent for? Why? Explain the notion of CAN and CAUN estimators. Give examples of the same with supporting proof. 4. (a) Define a consistent estimator. Give an example. State and prove that consistent estimators have an invariance property. Describe the general method of constructing the confidence intervals for large samples. Obtain 100(1 ) % confidence interval for the Poisson parameter in large samples. 5. (a) A sample of size 1 is taken from a Poisson distribution with parameter. To test H 0 : 1 against H 1 : 2. Consider a nonrandomised test ( x) 1 if x 3 and =0 if x 3. Find the probabilities of type I and types II errors and the power of the test. 2 (DMSTT 02)

6 Explain likelihood ratio test criterion and derive its asymptotic distribution. 6. (a) Define an UMP test. X is N (, ). Based an n independent observations on X derive the UMP test for testing 0 against : 2 0 when is known. 2 State and prove Neymann- Pearson lemma. Give its importance in testing the hypothesis. 7. (a) Explain the Mann-Whitney u test. Clearly mention the situations when it can be employed and also bring out its special uses. Explain kolomogorov-sminnor test. Make comparision of the test with chi-square goodness of fit test. 8. (a) Derive Wald SPRT of strength (, ) for testing H0 : 0 against H1 : 1 for a normal distribution with mean and unit variance. Explain the advantages of sequential method of testing of hypothesis. Describe wald's SPRT. 3 (DMSTT 02)

7 9. (a) Explain the SPRT for testing the Poisson parameter. Obtain OC and ASN functions. State and prove Wald's fundamental identity. 10. Write short notes on any TWO of the following : (a) Factorization theorem. Sufficient condition for consistency. (c) Randomized and non-randomized tests. (d) Wuilcoxan signed rank test. (e) Application of SPRT to binomial distribution. 4 (DMSTT 02)

8 (DMSTT 03) M.Sc. DEGREE EXAMINATION, DECEMBER First Year Statistics Paper III SAMPLING THEORY Time : Three hours Maximum : 100 marks Answer any FIVE questions. All questions carry equal marks. 1. (a) Discuss the main steps involved in a sample survey. What are the different sources of errors in a sample survey? Explain. 2. (a) Describe the organisation and functions of CSO. Discuss the basic principles of a sample survey. 3. (a) In srswor show that (i) sample mean and (ii) sample mean square are unbiased estimates of the corresponding population parameters. Explain the conditions under which stratification produce large gains in precision with suitable examples.

9 4. (a) Explain the purpose of stratification in sample surveys. Compare the efficiencies of the Neymann and proportional allocations with that of unstratified random sample of the same size. Derive the Neyman's formula for optimum allocation. 5. (a) Explain systematic sampling. Obtain the variance of the mean based on systematic sample and compare the variance with that based on srs and stratified random sampling. Define cluster sampling. Derive the variance of the estimate of the mean per element in cluster sampling with equal sizes. 6. (a) Explain two stage sampling. Derive the variance of the estimated mean in two stage sampling. In sampling with ppswr obtain the estimator of the population total. Derive its variance. 2 (DMSTT 03)

10 7. (a) What is PPS sampling? Explain Lahiris' method of PPS sampling. In PPS sampling with replacement define an unbiased estimator of population mean. Explain the procedure of drawing a sample by two-stage sampling and give an example. Describe the applications of two-stage sampling. 8. (a) Distinguish between regression and ratio estimators in stratified sampling with examples. Show that K Yls wm ( Ym bm ( Xm xm ) m 1 represents the m th stratum. where m Distinguish between separate and combined regression estimators. Give estimates of population mean for these two parameters and compare their relative performance. 9. (a) Derive the bias and mean square error of the ratio estimator of the population total assuming srswor for the units. Discuss the relative efficiency of ratio and regression estimates. 3 (DMSTT 03)

11 10. Write short notes on any TWO of the following ; (a) Circular systematic sampling NSSO (c) Optimum cluster size (d) Multi stage sampling (e) Control of non-sampling errors. 4 (DMSTT 03)

12 (DMSTT 04) M.Sc. DEGREE EXAMINATION, DECEMBER First Year Statistics Paper IV DESIGN OF EXPERIMENTS Time : Three hours Maximum : 100 marks Answer any FIVE questions. All questions carry equal marks. 1. (a) Explain the role of Quadratic forms in design of experiments. State and prove Cochrans theorem. 2. (a) Explain the methodology of deriving Eigen valves and Eigen vector associated with a matrix. What is the importance of it? Define trace of a matrix. Explain its relationship in finding the derivative of a matrix with respect its elements.

13 3. (a) Explain the limitations ordinary least square estimators in linear models. Stating the assumptions clearly and bring out Gauss Monkov set up of a linear model. 4. (a) State and prove Alken's theorem. Explain the importance of G-inverve in estimating the parameters of a linear model. 5. (a) Explain the procedure of analysing a two way classified data. Discuss the uses of analysis and variance with two way classification. 6. (a) Derive the F-statistics for testing the equality K means in one way classified data. With the usual notation show that the degrees of freedom associated error in twoway classified ANOVA Test is ( n 1)( K 1). 7. (a) With the usual notations derive the least square estimates and expectations of sum of squares of a Randomised Block design. Describe the uses of R.B.D. 8. (a) Explain the layout of a L.S.D. with an example. Compare and contrast the Latin Square design with R.B.D. 2 (DMSTT 04)

14 9. (a) What is a treatment contrast? When one two such contrast are orthogonal? Show that in 2 3 experiment main effects and interaction effects are orthogonal. Explain the philosophy of Factorial experiments. 10. (a) State and prove Fisher's inequality in ab BD. When is ab BD is called symmetric? 3 (DMSTT 04)