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Statistics(Hons.) Semester II

Paper STH 201: Calculus-II

Review of integration and definite integrals, integration of irrational functions. Reduction formulae, application of integration: Rectification and quadrature, volumes and surfaces of revolution for cartesianand polar curves, differentiation under integral sign.
Double Integrals, change of order of integration, transformation of variables, Beta and Gamma integrals and relationship between them.

Geometry: Pair of straight lines, Circle, derivation of equation of tangent, normal, polar and lengthof tangent from any external point. Conic sections: Equation of Parabola and associated theorems, Ellipse, eccentric angle, equation of Ellipse and its tangents and normal in terms of eccentric angle, Hyperbola instandard forms and their properties, real, conjugate and rectangular Hyperbola.

SUGGESTED READINGS:

  • Gorakh Prasad and Gupta, H. C. (1994): Text Book on Coordinate Geometry, Pothishala ‘Pvt. Ltd., Allahabad.
  • Shanti Narain and Mittal, P.K. (2007): Integral Calculus. (Reprint). S. Chand and Co.
  • Strauss, M. J., Bradley, G. L. and Smith, K. J. (2007): Calculus (3rd Edition), Dorling Kindersley.(India) Pvt. Ltd. (Pearson Education).

Paper STH 202: Algebra-II

System of linear equations, row reduction and echelon forms, the matrix equations AX=B, solutionsets of linear equations, linear independence, Applications of linear equations, inverse of a matrix. Rank ofa matrix, row-rank, column-rank, standard theorems on ranks, rank of the sum and the product of twomatrices. Generalized inverse (concept with illustrations). Partitioning of matrices and simple properties. Homogeneous and non-homogeneous system of linear equations- their consistency and general solutions. Introduction to matrix polynomial. Characteristic roots and characteristic vectors of a matrix, CayleyHamilton theorem. Quadratic forms, linear orthogonal transformation and their diagonalisation.

Sets ,binary relations. Definitions and examples of groups,abelian-groups, rings, integral domain, skew-field and fields ,vector spaces with illustrations, vector space with real scalars, linear combination of vectors, sub-spaces, linear span, bases and change of bases, dimensions, orthogonal vectors, orthogonal basis, Gram-Schmidt orthogonalisation process. Matrix differentiation.

SUGGESTED READINGS:

  • Artin, M. (1994): Algebra. Prentice Hall of India.
  • Datta, K.B. ( 2002) : Matrix and Linear Algebra. Prentice Hall of India Pvt. Ltd.
  • 3. Graybill, F.E.(1961) :Introduction to Matrices with Applications in Statistics. Wadsworth Pub. Co. 4. Gupta, S.C. (2008): An Introduction to Matrices. (Reprint). Sultan Chand & Sons.
  • 5. Hadley, G. (2002) : Linear Algebra. Narosa Publishing House Reprint.
  • 6. Searle, S. R. (1982) : Matrix Algebra Useful for Statistics. John Wiley & Sons.

Paper STH 203: Probability and Statistical Methods-II

Random Variables: Discrete and continuous random variables, p.m.f. , p.d.f. , c.d.f. illustrations of random variables and its properties. Univariate transformations.Expectation of random variable and its properties. Moments and cumulants, moment generating function. Cumulant generation function and characteristic function.

Standard discrete probability distributions: Degenerate, Binomial, Poisson, Geometric, Negative Binomial, Hypergeometric. Standard continuous probability distributions: Normal, uniform, exponential, beta, gamma, Cauchy, Laplace.

SUGGESTED READINGS:

  • Goon, A.M., Gupta, M.K. and Dasgupta, B. (2003): An Outline of Statistical Theory, Vol. I, 4th Edn. World Press, Kolkata.
  • Gupta, S.C. and Kapoor, V.K. (2007): Fundamentals of Mathematical Statistics, 11th Edn., (Reprint), Sultan Chand and Sons.
  • Hogg, R.V. and Tanis, E.A. (2009): A Brief Course in Mathematical Statistics. Pearson Education.
  • Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994): Discrete Univariate Distributions, John Wiley.
  • Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994): Continuous Univariate Distributions, Vol. I &Vol. II, 2nd Edn., John Wiley.

Statistics(Hons.) Semester IV

Paper STH 401: Numerical Analysis

Numerical Analysis: Factorial with positive and negative index. Operators: Shift operator (E), ), average (µ), differential (D) ), central difference (forward difference (∆), backward difference (inter-relations between them. Finite differences of order n, divided differences of order n and interpolation. Newton’s forward, backward and divided difference interpolation formulae with error term. Lagrange’sinterpolation formula. Central difference formulae: Gauss and Stirling’s formulae.

Inverse interpolation: Lagrange’s inverse interpolation formula, Method of successive approximation and method of reversion of series. Summation of finite series. Numerical differentiation.
Numerical integration: Newton-Cote’s integration formula, Trapezoidal rule, Simpson’s one-third rule, Simpson’s three-eighth rule and Weddle’s rule with error term. Euler-Maclaurin’s summation formula. Stirling’s approximation to factorial n.
Solution of difference equations of first order with variable coefficients and linear difference equations with constant coefficients.

SUGGESTED READINGS:

  • Bradie, B. (2006): A friendly introduction to Numerical Analysis, Pearson Education,India.
  • Gerald, C. F. and Wheatly, P. O. (2005): Applied Numerical Analysis, Pearson Education,India.
  • Hilderbrand, F.B. (1974): Introduction to Numerical Analysis. Tata McGraw Hill.
  • Sastry, S.S. (2000): Introductory Methods of Numerical Analysis, 3rd edition, Prentice Hall of India Pvt. Ltd., New Delhi.
  • Saxena, H.C. (2005): Finite Differences and Numerical Analysis, 15th Revised Edn. (Reprint). S. Chand and Co.
  • Scarborough, J.B. (1966): Numerical Mathematical Analysis, 6th Edition. Oxford and IBH.

Paper STH 402: Probability and Statistical Methods- IV

Order Statistics: Introduction, distribution of rth order statistic, joint distribution of rth and sth order statistics.Sampling Distribution: Definitions of random sample, parameter and statistic, sampling distribution of a statistic, sampling distribution of sample mean, standard errors of sample mean and sample proportion. Sampling distributions of chi-square, t and F statistics. Distribution of sample correlation coefficient r when ρ = o.

Tests of significance: Null and alternative hypotheses, level of significance and probabilities of Type I and Type II errors, critical region and p-value. Large sample tests, use of CLT for testing single proportion, difference of two proportions, single mean, difference of two means, standard deviation and difference of standard deviations. Tests of significance based on t, F and Chi-square distributions.

SUGGESTED READINGS:

  • David, H.A. and Nagaraja, H.N. (2003): Order statistics, 3rd Edition, John Wiley and sons.
  • Goon, A.M., Gupta, M.K. and Dasgupta, B. (2003): An Outline of Statistical Theory, Vol. I, 4th Edn. World Press, Kolkata.
  • Gupta, S.C. and Kapoor, V.K. (2007): Fundamentals of Mathematical Statistics, 11th Edn., (Reprint). Sultan Chand and Sons.
  • Hogg, R.V. and Tanis, E.A. (2009): A Brief Course in Mathematical Statistics. Pearson Education.
  • Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994): Continuous Univariate Distributions, Vol. II, 2nd Edn. John Wiley.
  • Johnson, R.A. and Bhattacharya, G.K. (2001): Statistics-Principles and Methods, 4th Edn. John Wiley and Sons.

Paper STH 403: Operational Research

Introduction to OR: Phases of OR, model building and various types of OR Problems, Linear Programming: Models, graphical solution, simplex method and M-technique. Concept of duality, dual simplex method, post-optimality analysis.

The transportation problems: North-West corner rule, Least cost method, Vogel’s approximation method and MODI’s method to find the optimal solution. The assignment problem. Networking problems. Individual replacement model.

Game Theory: Rectangular games; methods of solution: dominance method, modified dominance, Graphical solution and algebraic technique solution by L.P. Method. Simulations: Simulation models, event-type simulation, generation of random phenomena, steps in simulation, application of simulation techniques.

SUGGESTED READINGS:

  • Gass, S.I. (1985): Linear Programming: Methods and Applications. Boyd Fraser Publishing Co., Danvers.
  • Hadley, G. (2002): Linear Programming (Reprint). Narosa Publishing House.
  • Hillier, F.S. and Lieberman, G. J. (2001): Introduction to Operational Research, 7th Edn. Irwin.
  • Kantiswarup, Gupta, P.K. and Manmohan (2008): Operations Research, 13th Edn. Sultan Chand and Sons.
  • Sharma, S.D. (2009): Operations Research-Theory, Methods and Applications, 16th Revised Edn., Kedar Nath Ram Nath.
  • Taha, H.A. (2007): Operations Research: An Introduction, 8th Edn. Prentice Hall of India.

Statistics(Hons.) Semester VI

Paper STH 601: Statistical Inference-II

Testing of Hypothesis: Statistical hypothesis, simple and composite hypotheses. Test of statistical hypotheses, null and alternative hypotheses. Critical region. Two kinds of errors. Level of significance and power of a test. Consistency and relative efficiency of tests. MP test and region. Neyman-Pearson Lemma, critical regions for simple hypotheses, for one parameter. Randomized test. UMPU Test and region. Likelihood ratio test, properties of LR tests (without proof). Sequential Probability Ratio Test. Determination of stopping bounds A and B, OC and ASN functions of SPRT. Non-Parametric tests. Empirical distribution function, one sample and two-sample sign test. Wald-Wolfowitz run test. Run test for randomness, Median test, Wilcoxon-Mann-Whitney U-test. Kolmogorov-Smirnov one-sample test, Kruskal-Wallis test.

SUGGESTED READINGS:

  • Dudewicz, E.J., and Mishra, S.N. (1988): Modern Mathematical Statistics, John Wiley & Sons.
  • Gibbons, J. D. and Chakraborty, S. (2003): Non parametric Statistical Inference, 4th Edition, Marcel Dekker, CRC.
  • Goon, A.M., Gupta, M.K. and Dasgupta, B. (2005): An Outline of Statistical Theory, Vol. II, 3rd Edn. World Press, Kolkata.
  • Hogg, R.V. and Tanis, E.A. (1988): Probability and statistical inference, 3rd Edn. Macmillan Publishing Co., Inc.
  • Rohatgi, V. K. and Saleh, A. K. Md. E. (2009): An Introduction to Probability and Statistics, 2nd Edn. (Reprint). John Wiley and Sons.
  • Sigel, S. (1956): Nonparametric Statistics for the Behavioural Sciences. McGraw Hill, N.Y.

Paper STH 602: Design of Experiments

Experimental designs: Role, historical perspective, terminology, experimental error, basic principles, uniformity trials, fertility contour maps, choice of size and shape of plots and blocks. Basic designs: Completely Randomized Design (CRD), Randomized Block Design (RBD), Latin Square Design (LSD) – layout, model and statistical analysis, relative efficiency, analysis with missing observations.

Incomplete Block Designs: Balanced Incomplete Block Design (BIBD) – parameters, relationships among its parameters, incidence matrix and its properties, Symmetric BIBD, Resolvable BIBD, Affine Resolvable BIBD, Intra Block analysis, complimentary BIBD, Residual BIBD, Dual BIBD, Derived BIBD. Fractional factorial experiments: Construction of one-half and one-quarter fractions of 2n (n≤5) factorial experiments, Alias structure, Resolution of a design.

SUGGESTED READINGS:

  • Cochran, W.G. and Cox, G.M. (1959): Experimental Design. Asis Publishing House.
  • Das, M.N. and Giri, N.C. (1986): Design and Analysis of Experiments. Wiley Eastern Ltd.
  • Goon, A.M., Gupta, M.K. and Dasgupta, B. (2005): Fundamentals of Statistics. Vol. II, 8th Edn. World Press, Kolkata.
  • Kempthorne, O. (1965): The Design and Analysis of Experiments. John Wiley.
  • Montgomery, D. C. (2008): Design and Analysis of Experiments, John Wiley.

Paper STH 603: Econometrics

Objectives behind econometric models, General Linear Model: assumptions, least-squares estimation. BLUE, analysis of variance, tests of significance, confidence intervals for the prarameters, Prediction, Estimation under linear restrictions. Multicollinearity, concept, detection of multicollinearity, consequences and solutions of multicollinearity. Generalized least squares estimation, Aitken estimators, Heteroscedastic disturbances, efficiency of Aitken estimator with OLS estimator under heteroscedasticity. Autocorrelation: concept, consequences of autocorrelated disturbances, detection of autocorrelation, their estimation and testing, estimation using Durbin-Watson statistic. Forecasting: exponential smoothing for linear trend model.

SUGGESTED READINGS:

  • Draper, N.R. and Smith, H. (1998): Applied Regression Analysis, 3rd Edn. John Wiley and Sons.
  • Gujarati, D.N. (2006): Essentials of Econometrics, 3rd Edn., McGraw-Hill.
  • Johnston, J. (1991): Econometric Methods, 3rd Edn. McGraw-Hill Kogakusha Ltd.
  • Koutsoyiannis, A. (1984): Theory of Econometrics: An Introductory Exposition of Econometric Methods, 2nd Edn. Macmillan.
  • Maddala, G.S. (2002): Introduction to Econometrics, 3rd Edn. John Wiley and Sons.