ASSIGNMENT - 1, MAY M.Sc. (PREVIOUS) FIRST YEAR DEGREE STATISTICS. Maximum : 20 MARKS Answer ALL questions.

Transcription

1 (DMSTT 0 NR) ASSIGNMENT -, MAY-04. PAPER- I : PROBABILITY AND DISTRIBUTION THEORY ) a) State and prove Borel-cantelli lemma b) Let (x, y) be jointly distributed with density 4 y(+ x) f( x, y) = y(+ x) e, x, y 0 = 0, otherwise Find E{X/Y) and E(Y/X). ) a) State and prove Minkowski inequality. b) Define characteristic function. State its properties. x, x,... xn be random variables with joint p.d.f f and let fj be the marginal p.d.f of xj, j =,,...n. State and prove a necessary and sufficient condition for x, x,...xn to be independent. 3) a) State and prove Kolomogrov s strong law of large numbers for independent random variables. b) State and prove Khintchine s form of weak law of large numbers. 4) a) Investigate a.s convergence of {Xn} to 0 with P(Xn = 0) = n P(Xn = ± ) = n b) State and prove Liapounov s form of central limit theorem. 5) a) Define multinomial distribution. State and prove the property that characterizes the multinomial distribution. b) Define compound binomial. Obtain its mean and variance.

2 (DMSTT 0 NR) ASSIGNMENT -, MAY-04. PAPER- I : PROBABILITY AND DISTRIBUTION THEORY ) a) Define compound Poisson. Obtain its m.g.f. Hence find its mean and variance. b) Define a trinomial distribution. Obtain its m.g.f. ) a) Define Laplace distribution. Obtain its m.g.f. Hence find its mean and variance b) Define Weibull distribution. Obtain its m.g.f. Hence find its variance 3) a) Define Logistic distribution. Obtain its m.g.f. Hence find its variance b) Define Lognormal distribution. Find its mean and variance. 4) a) Define F-statistic. Derive its distribution. b) In the usual notation. Obtain the joint p.d.f of X() X()...X(n). 5) a) Obtain the joint p.d.f of X(j) and X(k), j < k n. b) Define t-statistic. Derive the p.d.f. of non-central t-distribution.

3 (DMSTT 0 NR) ASSIGNMENT -, MAY-04. PAPER- II : STATISTICAL INFERENCE ) a) Define minimal sufficient statistic. Give an example Let x i (i =,,.., n) be a random sample from an exponential distribution with p.d.f. f ( x) = exp[ ( x θ)] if θ <x< θ = 0 otherwise. Where < θ <, Obtain the sufficient statistic for θ. b) State and prove Cramer - Rao inequality ) a) Let x, x,.., x n be a random sample from N ( µ, σ ) where µ and σ are both unknown. Obtain a sufficient statistic for θ = ( µ, σ ). Show that it is a complete statistic. b) State and prove Lehmann-Scheffe theorem 3) a) Define a consistent estimator. State and prove the sufficient conditions for consistency. b) Define CAN and CUAN estimators construct a 99% confidence interval for difference of means in large samples. 4) a) For a random sample of size n from a normal distribution with unknown mean µ show that the sample mean is an efficient estimator. Show that it is also CUAN estimator b) Explain the maximum likelihood method. Show that in sampling from the distribution with p.d.f. f ( x) = θexp[ θx], 0 < x < = 0 otherwise, is the m.l. estimator of θ and has a greater X θ variance than the unbiased estimator n n X. 5) a) Explain the relation between testing and interval estimation. Let x, x,.., x n. be a random sample from a Poisson distribution with an unknown parameter θ If n = 5 show that no MP non-randomised test exists for testing H o : θ = 3 against H : θ = θ (>3) Obtain an M.P. randomised test for testing H o. b) State and prove Neyman-Pearson lemma.

4 (DMSTT 0 NR) ASSIGNMENT -, MAY-04. PAPER- II : STATISTICAL INFERENCE ) a) Explain likelihood ratio test. Discuss its properties. b) Let X, X,. X n be a random sample from N ( µσ ) where µ and σ are both unknown. Derive the likelihood ratio test for testing H o : µ = µ o against H : µ > µ o. ) a) Explain : (i) Sign test and (ii) Kolmogorov-Smirnov test. b) Explain Mann-whitney test 3) a) Explain : (i) Run test and (ii) Wilcoxon signed rank test. b) Fifteen 3-year-old and fifteen 3-year-old girls were observed during two sessions of recess in a nursery school. Each Child s play was scored for incidence and degree of agression as follows : Boys : 96, 65, 74, 78, 8,, 68, 79,, 48, 53, 9, 8, 3, 40 Girls:, 47, 3, 59, 83, 4, 3, 5, 7, 8,, 34, 9, 5, 5 Is there evidence to suggest that there are gender differences in the incidence and amount of aggression? Use Mann-whitney-wilcoxon test. 4) a) Explain SPRT procedure. Show that it terminates with probability one. b) Let X be a random variable having the p.m.f. x x ( x) = θ ( θ) fθ if x = 0, ; = 0 otherwise. Develop an SPRT procedure for testing H o : θ = θo against H: θ = θ. Obtain the O.C and ASN functions of the test. 5) a) Explain SPRT procedure. Derive its OC and ASN functions. b) Let X, X,. X n be i.i.d random variables from N ( θ, σ ) with unknown θ and known Derive the SPRT to test H o : θ = against H : θ = θ. Obtain its OC and ASN functions. θo σ

5 (DMSTT 03 NR) ASSIGNMENT -, MAY-04. PAPER- III : SAMPLING THEORY ) a) What are the advantages of the sampling method? Explain the uses of sample survey. What is a sampling frame? b) Describe the structure and functions of C.S.O. ) a) Explain sampling and non-sampling errors. Explain the uses of random number tables. b) Explain the principal steps in a sample survey. 3) a) Explain simple random sampling with and without replacements. Obtain the variance of an estimate of the population total in srswor. b) Discuss the gain in precision due to stratification. 4) a) Assuming srs determine the sample size in sampling i) for proportions and ii) continuous data. Obtain the variance of the estimate of mean in stratified sampling. b) Explain proportional allocation and optimum allocation. Compare their precision with simple random sampling. 5) a) Explain systematic sampling and circular systematic sampling. Give their applications two each. Explain cluster sampling with a suitable example. b) Determine the optimum cluster size for fixed cost.

6 (DMSTT 03 NR) ASSIGNMENT -, MAY-04. PAPER- III : SAMPLING THEORY ) a) Obtain the variance of an unbiased estimate of the mean per unit in cluster sampling. b) Obtain the variance of the estimated mean in systematic sampling in terms of intra class correlation ( st coefficient. Compare it with the V y ) in populations with linear trend. ) a) Explain Hansen and Huniitz and Lahiri s Methods of selecting pps sample. Obtain an unbiased estimate of the population total. b) Explain multistage sampling with a suitable example. In two stage sampling obtain an estimate of variance of estimate of population mean per unit 3) a) Obtain the variance of an estimate of population total in pps sampling. b) Explain two stages sampling. What are its advantages? Obtain an unbiased estimate of variance of sample mean per subunit. 4) a) Define : i) ratio estimate of the population ratio and ii) ratio estimate of the population total. Obtain the estimate of y ). Explain the ratio estimates in stratified sampling. V( st b) Obtain V y ). Compare it with the variance of ratio estimate. ( st 5) a) Explain the regression estimates in stratified sampling. b) Show that the ratio estimator is optimum under the conditions to be stated by you.

7 ASSIGNMENT -, MAY-04. PAPER- IV : DESIGN OF EXPERIMENTS (DMSTT 04 NR) ) a) Define : (i) rank of a matrix (ii) orthogonal matrix (iii) idempotent matrix and give examples one each. Explain the concept of differentiation using matrices. b) State and prove Cayley-Hamilton theorem. ) a) Find the characteristics roots and the characteristic spaces for the following matrix : b) State Cochran s theorem. Find the inverse of the following matrix: ) a) Explain : i) Linear model and ii) Best linear unbiased estimates b) Explain general linear model. State and prove Ailken s theorem 4) a) Explain : (i) Gauss-Markov set up (ii) General linear model and (iii) estimability. State and prove a necessary and sufficient condition for the estimability of parametric functions. b) State and prove Gauss-Markov theorem. 5) a) Explain the ANOVA of two-way classification with unequal number of observations. b) Explain the ANCOVA with a single concomitant variable.

8 (DMSTT 04 NR) ASSIGNMENT -, MAY-04. PAPER- IV : DESIGN OF EXPERIMENTS ) a) Explain ANOVA of three-way classification with unequal number of observations. b) Explain the ANCOVA of two-way classification. ) a) What is meant by design of an experiment? Why is randomisation necessary? Explain the missing plot technique. b) Explain the analysis of LSD. Obtain its efficiency relative to RBD. 3) a) Explain the analysis of RBD with one missing observation. b) Define Graeco Latin Square Design. Explain its analysis. Obtain its efficiency relative to RBD. 4) a) What are factorial experiments? What are their advantages? Explain the analysis of 3 factorial experiment. b) Explain factorial experiment. Explain Yates method in factorial experiment. 5) a) Explain 3 3 factorial experiment. Explain Yates method in 3 factorial experiment. b) Explain BIBD with a suitable example. Explain interblock analysis of BIBD.