5) a)define probability generating function. State and prove any two properties of probability generating function.

Transcription

1 (DMSTT 01 NR) ASSIGNMENT - 1, DEC PAPER- I : PROBABILITY AND DISTRIBUTION THEORY 1) a)define distribution function and list its properties. b) State and prove Borel-Cantelli Lemma. 2) a)state and prove inversion theorem of characteristic function. b) Derive Minikowski inequality. 3) a)discuss the different types of convergence of sequence of random variables. b) State and prove Kolmogrov s storng law of large numbers. 4) a)state and prove Levy and Lindeberg form of central limit theorem. b) State and prove Liapounov s form of Central limit theorem. 5) a)define probability generating function. State and prove any two properties of probability generating function. b) Define characteristic function and list out its properties.

2 (DMSTT 01 NR) ASSIGNMENT - 2, DEC PAPER- I : PROBABILITY AND DISTRIBUTION THEORY 1) a) Let X be a random variable. Let EX k for some k > 0. Then prove that n k P X n 0 as n. b) Write about compound poisson. 2) a)define Multinomial distribution obtain its m.g.f. b) Deduce mean and variance of log normed distribution. 3) a)deduce moment generating function of Laplace distribution. b) Define rectangular and deduce m.g.f. to rectangular distribution. 4) a)define t-distribution. Deduce properties of t-distribution. b) Probability density function of joint probability density function of two order statistics. 5) a)define non central F-distribution. b) Derive the density of non-central χ 2 - distribution.

3 (DMSTT 02 NR) ASSIGNMENT - 1, DEC PAPER- II : STATISTICAL INFERENCE 1) a)state and prove Fisher - Neymann factorization Theorem. b) State and prove Cramer - Rao inequality. 2) a)state and prove Rao-Black well Theorem. b) Define unbiasedness and consistency. Show that unbiased estimator is not unique. 3) a)find UMVUE of the mean of a Normal distribution whose standard deviation is one. b) State Lehman Scheffe theorem. Give its importance. 4) a)explain the principle of maximum likelihood estimation. b) Explain CAN and CAUN estimators and detail its properties. 5) a)state and prove Neymann-Pearson lemma and bring art the use of critical function of most powerful tests. b) Explain Monotone likelihood Ratio criterion by example using normal distribution.

4 (DMSTT 02 NR) ASSIGNMENT - 2, DEC PAPER- II : STATISTICAL INFERENCE 1) a) Define Randomised and non-randomised test procedures and give the examples of the same. b) Derive UMP test for testing for simple Hypothesis about variance of a normal population. 2) a)bring out the differences between parametric and Non-parametric tests. b) Explain the procedure of signtest. 3) a)describe the Kolmogrov Smirnov two sample test. b) Explain the concept of a run in the case of testing randomness in a series of observations. 4) a)describe briefly the SPRT procedure. b) Define OC and ASN functions of the SPRT. 5) a)show that SPRT terminates eventually. b) Describe SPRT in the case of a Binomial distribution.

5 (DMSTT 03 NR) ASSIGNMENT - 1, DEC PAPER- III : SAMPLING THEORY 1) Define: a) Population. b) Sampling frame. c) Sampling scheme. d) Sampling design. e) Complete enumeration Survey. f) Sampling strategy. 2) a)write about Sampling errors and give its sources. b) Explain about NSSO. 3) a)deduce mean and variance of SRSWOR. b) Explain about the optimum allocation. 4) a)distinguish between Simple Random Sampling and systematic Random Sampling. b) Compare SRS with proportional allocation and optimum allocation. 5) a)define cluster sampling. Deduce mean and variance. b) Describe the method of cluster sampling with equal sized clusters.

6 (DMSTT 03 NR) ASSIGNMENT - 2, DEC PAPER- III : SAMPLING THEORY 1) a)explain the linear and circular systematic sampling procedures with suitable examples. b) Explain PPS sampling with and without replacement. 2) a)define Horvitz Thompson estimator. Derive unbiased estimator of the variance of Horvitz Thompson estimator. b) Write about two stage sampling. Deduce mean and variance of two-stage sampling. 3) a) Write about Murthy s Method. b) Write about Brewer s Method. 4) a)describe the Ratio method of estimation. Find its expected value and variance. b) Write about the simple regression estimate of the population mean is it unbiased? If not, when it is unbiased. 5) a)compare Regression, Ratio and SRSWOR methods. b) Explain Double Sampling for Regression estimation and its uses.

7 (DMSTT 04 NR) ASSIGNMENT - 1, DEC PAPER- IV : DESIGN OF EXPERIMENTS 1) a)define definition of marks and list out their properties. b) Find rank, inverse, orthogonal of a matrix ) a)state and prove Cayley - Hamilton Theorem b) Find inverse of the matrix A = ) a)explain general linear model and give its properties. b) Explain about best linear unbiased estimates and explain how this is useful in ANOVA. 4) a)state and prove Gauss-Markoff theorem. b) Explain estimability of linear parametric function. 5) a)what are the underlying assumptions of ANOVA model. b) Explain ANOVA of two-way classification with unequal number of observations.

8 (DMSTT 04 NR) ASSIGNMENT - 2, DEC PAPER- IV : DESIGN OF EXPERIMENTS 1) a) Explain Analysis of Covariance of one-way classification. b) Explain ANOVA of three way classification with equal number of observations. 2) a)explain the procedure for constructing orthogonal latin square design. b) Explain Graeco-Latin square design. Give its analysis. 3) a) Show that in 2 3 factorial experiment the main effects and interaction effects are mutually orthogonal. b) What is factorial experiment? Explain its importance. 4) a)explain analysis of RBD obtain LSD relative efficiency to RBD. b) Write about BIBD of 3 3 factorial experiment. 5) a)write about analysis of LSD with single missing observation. b) Explain Yetes method in 2 3 factorial experiment.