COURSE STRUCTURE OF INT. M. SC. STATISTICS. Integrated M. Sc. I to VI Semester Sem. Paper Code Title Credit

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1 COURSE STRUCTURE OF INT. M. SC. STATISTICS Integrated M. Sc. I to VI Semester Sem. Paper Code Title Credit I II III IV V VI BST 101 Analysis of Univariate Data & Elements of Probability BST 102 Practical based on BST BST 103 Analysis of Bivariate Data & Random Variables BST 104 Practical based on BST BST 201 Probability Distributions I BST 202 Practicals based on BST BST 203 Statistical Inference I BST 204 Practicals based on BST BST 301 Probability Distributions -II BST 302 Sample Survey BST 303 Time Series & Index Numbers BST 304 Statistical Inference II BST 305 Numerical Analysis BST 306 Practicals BST 351 Elements of Reliability & Survival Analysis BST 352 Design of Experiments-I BST 353 Statistical Process Control & Vital Statistics BST 354 Operation Research BST 355 Official Statistics & Demand Analysis BST 356 Practicals Total Theory Tutorial Practical Integrated M. Sc. VII Semester Paper Code Title Credit Credit MST 101 Probability Theory MST 102 Distribution Theory MST 103 Linear Algebra and Matrix Theory MST 104 Sampling Theory MST 105 R: Statistical Programming Language MST 106 Practicals

2 Integrated M. Sc. VIII Semester Paper Code Title Credit Credit MST 201 Estimation and Testing of Hypotheses MST 202 Linear Models MST 203 Stochastic Models MST 206 Practicals Elective Courses Paper Code Title Prerequisite Credit Credit MST 221 Design of experiment II MST 222 Econometrics MST 223 Financial Mathematics MST 224 Economics for Actuarial Science MST 225 Machine Learning MST 226 Modeling and Simulation (Mathematics, MTM 204) MST 227 Mathematical Software and Tools (Mathematics, MTM 206) Integrated M. Sc. IX Semester Paper Code Title Pre-requisite Credit Credit MST 301 Decision Theory and Non Parametric Inference MST 302 Time Series Analysis and Forecasting MST 303 Multivariate Analysis MST 306 Practicals Elective Courses Paper Code Title Pre-requisite Credit Credit MST 321 Statistical Quality Control MST 322 Stochastic Finance MST 323 Contingencies MST 324 Principle and Practices of Insurance MST 325 Data Mining

3 Integrated M. Sc. X Semester Paper Code Title Pre-requisite Credit Credit MST 401 National Development Statistics MST 406 Practicals MST 410 Project Elective Courses Paper Code Title Pre-requisite Credit Credit MST 421 Survival Models and Analysis MST 422 Advanced Bayesian Inference MST 423 Statistical Quality Management MST 424 Statistical Methods for Non-life Insurance MST 425 Life and Health Insurance MST 426 Employee Benefit Plans MST 427 Valuation and Loss Reserving MST 428 Finance and Financial Reporting

4 PAPER CODE BST 101 Analysis of Univariate Data & Elements of Probability TOTAL HOURS 43 Unit-1 Lectures: 15 Meaning and scope of the word Statistics. Data types: Qualitative and Quantitative Data scales of measurements: nominal, ordinal, ratio, interval Representation: Tabulation Compilation, Classification. Graphical and diagrammatic representation: Bar diagrams, multiple and stack bar diagrams, Histogram, Frequency Polygon, Frequency Curve, Ogive, Pie diagram, Box plot, Stem and leaf diagrams. Measures of Central Tendency: Concept, requirements of a good measure. Arithmetic Mean (A.M), Geometric Mean (G.M), Harmonic Mean (H.M.), Median, Mode: properties, merits and demerits. Quartiles, Deciles and Percentiles, Graphical method of determination of Median, Mode and Quantiles. Unit-2 Lectures: 08 Measures of Dispersion: Concept, Requirements of a good measure of dispersion. Range: Quartile Deviation (Semi-interquartile range): Coefficient of Q.D. Mean Deviation (M. D.), Proof of Minimal property of M.D. Mean Square Deviation (M.S.D.): proof of Minimal property of M.S.D. Variance and Standard Deviation (S.D): Effect of change of origin and scale, S.D. of pooled data (proof for two roups), Coefficient of Variation (CV). Moments: Raw moments and Central moments, relation between central moments and raw moments, Sheppard correction for moments (without derivation), Skewness: Measure of skewness, Types of skewness, Kurtosis, Types of kurtosis, Measure of kurtosis. Unit-3 Lectures: 10 Concepts of experiments: deterministic, probabilistic, outcomes of experiments. Sample space, Discrete (finite and countably infinite) and continuous sample space, Event, Elementary event, Compound event. Algebra of events (Union, Intersection, Complementation), De Morgan s law. Definitions of Mutually exclusive events, Exhaustive events, Venn diagram. Definition; Axiomatic definition of probability; Addition theorem (Proof of the result up to three events), Elementary properties, Classical definition of Probability as a special case, Probability as an approximation to the relative frequency, illustrative examples for computation of events based on Permutations and Combinations, with and without replacements, impossible events, certain events. Unit-4 Lectures: 10 Definition of conditional probability of an event, Multiplication theorem for two events, Independence of events: Pairwise and Mutual Independence of events. Partition of sample space. Statement and proof of Bayes theorem. 1. Rohatgi V. K. and Saleh A. K. Md. E., An Introduction to probability and Statistics. John Wiley & Sons (Asia). 2. Mukhopadhyay, P., Mathematical Statistics, new Central Book Agency Pvt. Ltd., Calcutta. 3. Hoel P. G., Introduction to Mathematical Statistics, Asia Publishing House. 4. Meyer P. L., Introductory Probability and Statistical Applications, Addision Wesley. 5. AM Goon, M K Gupta and B. Das Gupta, Fundamentals of Statistics, Volume-I, World Press. 4

5 PAPER CODE BST 102 Practical based on BST 101 CREDIT 01 (0-0-1) CONTENT Students will be required to do parcticals, based on topics listed below, using MS Excel: 1. Introduction to MS Excel: Data storage, elementary calculations and graphical representations. 2. Actual conduct of experiments and comparing relative frequencies with actual probabilities. 3. Computation of conditional probabilities (Bayes theorem) 4. Tabulation and Construction of frequency distribution 5. Diagrammatic (Multiple stack bar diagrams, histogram, stem and leaf, pie chart) and graphical (frequency polygon, frequency curve) presentation of the frequency distribution. 6. Measures of Central tendency I (ungrouped data). 7. Measures of Central tendency II (grouped data). 8. Measures of Central tendency III (pooled data). 9. Computation of quantiles by use of Ogive curves, 10. Measures of the Dispersion I (ungrouped data). 11. Measures of the Dispersion II (grouped data). 12. Moments, Skewness & Kurtosis-I (ungrouped data). 13. Moments, Skewness & Kurtosis-II (grouped data). 14. Computation of raw, central moments, Pearson s coefficient of skewness and kurtosis. 5

6 PAPER CODE BST 103 Analysis of Bivariate Data & Random Variables TOTAL HOURS 44 Unit-2 Lectures: 8 Bivariate Data. Scatter diagram. The concept of dependency, illustrative real life examples. Covariance: Definition, Effect of change of origin and scale. Karl Pearson s coefficient of correlation (r): Definition, Properties, Spearman s rank correlation coefficient: Definition, Interpretation when r = -1, 0, 1, Derivation of the formula for without ties and Modification of the formula for with-ties Computation, variance of linear combination of variables. Unit-2 Lectures: 8 Concept of regression, Lines of regression, Fitting of lines of regression by the least square method. Regression coefficients (b xy, b yx) and their geometric interpretations, Properties. Derivation of the point of intersection of two regression lines and the acute angle between the two lines of regression. Unit-3 Lectures: 14 Definition of random variable, Discrete and continuous and mixed type of random variables, Definition of distribution function, Distributions function (df) of random variable, Probability distribution of function of random variable. Probability mass function (p.m.f.) and cumulative distribution function (c.d.f.) of a discrete random variable, Probability density function (p.d.f.) and cumulative distribution function (c.d.f.) of a continuous random variable, relation between df and pmf/pdf, Median and Mode of a univariate discrete and continuous random variables. Unit-4 Lectures: 14 Definition of expectation of a random variable, expectation of a function of a random variable, simple properties, Definitions of mean, variance of univariate distributions, Effect of change of origin and scale on mean and variance, Definition of raw, central moments, mean deviation. Pearson s coefficient of skewness, kurtosis, Definitions probability generating function (p.g.f.), moment generating function (m.g.f.)and characteristic function of a random variable, Effects of change of origin and scale. p.g.f. of sum of two independent random variables is the product of p.g.f.s (statement only), Derivation of mean and variance by using p.g.f. 1. Mood A. M., Grabyll R. A. and Boes D. C., Introduction to the theory of Statistics, Tata MeGraw Hill 2. Mukhopadhyay, P., Mathematical Statistics, new Central Book Agency Pvt. Ltd., Calcutta. 3. AM Goon, M K Gupta and B. Das Gupta, Fundamentals of Statistics, Volume-I, World Press. 4. Ross Sheldon M., Introduction to Probability Models, Academic Press 5. Rao, B. L. S. Prakash, A first course in probability and Statistics, World Scientific. 6

7 PAPER CODE BST 104 Practical based on BST 103 CREDIT 01 (0-0-1) CONTENTS Students will be required to do practicals, based on topics listed below, using MS Excel: 1. Advanced level commands /programming using MSEXCEL 2. Scatter diagram for bivariate data and interoperation. 3. Correlation coefficient: grouped data. 4. Correlation coefficient: ungrouped data 5. Spearman s Rank correlation: ungrouped data. 6. Regression-I (ungrouped data). 7. Regression-II (grouped data). 7

8 PAPER CODE BST 201 Probability Distributions I CREDIT 04 (3-1-0) Unit-1 Lectures: 8 General concept of a finite discrete random variable, computation of their means and variances, Concept of discrete probability distributions. Uniform, Bernoulli, Binomial, Poisson and geometric distributions with their properties and applications. Unit-2 Lectures: 8 Definition of continuous random variable (r.v.), illustrations, probability density function (p.d.f.), and cumulative distribution function (c.d.f.), properties of c.d.f., concept of probability distribution. Uniform, exponential, Beta (I and II kind) and Normal distributions with their properties and applications. Normal distribution as limiting case of binomial and Poisson distribution (without proof). Unit-3 Lectures: 10 Transformation of univariate continuous and discrete r.v.s: Distribution of Y=g(X), where g is monotonic using (i) Jacobian of transformation (ii) Distribution function and (iii) m.g.f. Unit-4 Lectures: 10 Meaning of random sample from a distribution, statistic, standard error, computation of sampling distribution of sum of random variables(discrete and continuous), CLT for iid (statement), Distribution of mean and variance from normal population. 1. Rohatgi V. K. and Saleh A. K. Md. E., An Introduction to probability and Statistics. John Wiley & Sons (Asia). 1. Hogg R.V. and Criag A.T.: Introduction to Mathematical Statistics (Third edition), Macmillan Publishing, New York. 2. Walpole R.E. & Mayer R.H.: Probability & Statistics, MacMillan Publishing Co. Inc, New York 3. Mayer P.L.: Introductory probability & Statistical Applications. Addison Weseley Publication Co., London. 4. Goon A.M., Gupta A.K. and Dasgupta B.: Fundamentals of Statistics (Vol. II) World Press, Calcutta. 5. Mukhopadhyay P. (1996): Mathematical Statistics, New central Book Agency (P) Ltd. Calcutta. 8

9 PAPER CODE BST 202 Practicals based on BST 201 CREDIT 01 (0-0-1) Total hours 45 CONTENT Students will be required to do practicals, based on topics listed below, using MS Excel: 1. Sketching of p.m.f., d.f. of Binomial and Poisson distributions 2. Sketching of p.d.f., d.f. of exponential, Beta (I and II kind) and Normal distributions distributions 3. Fitting/Application of Binomial and Poisson distributions 4. Fitting/Application of exponential, Beta (I and II kind) and Normal distributions 5. Computation of Mean, Median, Mode, Range, Quartile, Interquartile Range and Variance for the above distributions 6. Computation of probabilities corresponding to Uniform, Exponential, Beta, Normal Distribution 7. Simulation of data from discrete and continuous distributions 8. Sampling Distribution of mean and Variance 9. Distribution of Y=g(X) 9

10 PAPER CODE BST 203 Statistical Inference I CREDIT 04 (3-1-0) TOTAL HOURS 44 Unit-1 Lectures: 12 Concept of Statistical inference, sampling method and complete enumeration, Definition of population, parameter, parameter space, problem of estimation: point, intervals and testing of hypotheses. Definitions of an estimator, mean squared error (MSE) of an estimator, comparison of estimators based on MSE function. Unbiasedness: Unbiased estimator, Illustration of unbiased estimator for the parameter and parametric function. Definitions of Consistency, Sufficient condition for consistency, concept of efficiency and sufficiency. Neyman- Factorization theorem (without proof) Unit-2 Lectures: 8 Methods of estimation: Methods of moments, concept of likelihood function, Maximum Likelihood, Properties of MLE (without proof), Estimation of the parameters of normal distribution and other standard distributions by MLE. Unit-3 Lectures: 14 Hypothesis, types of hypothesis, problems of testing of hypothesis, critical region, type I and type II errors, probabilities of type I & type II errors. Power of a test, observed level of significance, p-value, size of a test, level of significance, testing for the mean and variance of Univariate normal population, testing of equalities of two means and variances of two Univariate normal populations. Definition of Most Powerful (MP) test, Neyman - Pearson (NP) lemma for simple null hypothesis against simple alternative hypothesis (proof), Examples of construction of MP test of level α, Testing hypotheses related to parameters of normal distribution including when both the parameters are unknown, Power curve of a test. Unit-4 Lectures: 10 Notion of non-parametric statistical inference (test) and its comparison with parametric statistical inference, Run test for one and two independent sample problems. Sign test for one sample and two sample paired observations, Wilcoxon's signed rank test for one sample and two sample paired observations, Mann-Whitney U - test (two independent samples), Kolmogorov Smirnov test for one and for two independent samples. 1. George Casella, Roger L. Berger (2002), Statistical Inference, 2 nd ed., Thomson Learning. 2. Mukhopadhyay P. (1996): Mathematical Statistics, New central Book Agency (P) Ltd. Calcutta. 3. Rohatgi, V.K. (1984): An Introduction to Probability Theory and Mathematical Statistics, Wiley Eastern. 4. Goon, Gupta & Das Gupta (1991): An Outline of Statistical Theory, Vol. II, World Press. 5. Hogg, R.V. and Craig, A.T. (1971): Introduction to Mathematical Statistics, McMillan. 10

11 PAPER CODE BST 204 Practicals based on BST 203 CREDIT 01(0-0-1) CONTENT Students will be required to do practicals, based on topics listed below, using MS Excel: 1. Point estimation by Method of moments and maximum. 2. Mean squared error and unbiasedness of an estimator 3. Sketching of power curve 4. Testing of simple hypothesis against simple alternative hypothesis (Finite valued discrete distributions, mean and variance of normal population). 5. Sign test for one sample and two samples 6. Wilcoxon's signed rank test for one sample and two samples 7. Run test (for one and two independent samples) and Sign test and Wilcoxon s signed rank test (for one and two samples paired observation). 8. Mann-whitney U- test (for two independent samples) and Median test (for two large independent samples) 9. Kolmogorov-Smirnov test (for one and two independent samples). 11

12 PAPER CODE BST 301 Probability Distributions -II CREDIT 04 (3-1-0) Unit-1 Lectures: 12 Exact sampling distributions: Chi square distribution, Student s t- distribution and Snedecor s F distribution. Definitions, derivation of p.d.fs, sketch of p.d.fs. for various values of parameter, moments. Inter relation between t, F and χ2 (without proof). Applications of t, F and χ2 distributions. Laplace (Double Exponential) Distribution, Gamma distribution, Lognormal Distribution and Cauchy Distribution. P. d. f., Nature of the probability curve, Moments. Unit-2 Lectures: 10 Negative Binomial Distribution: p.m.f., mean, variance, m.g.f., geometric distribution is a particular case of Negative Binomial distribution., Recurrence relation for successive probabilities. Limiting property of the Negative Binomial Distribution. Hypergeometric Distribution: p.m.f., Computation of probabilities of different events, mean, variance, Recurrence relation for successive probabilities, Binomial approximation to Hypergeometric. Multinomial distribution: P.m.f., marginal distributions, m.g.f. Unit-3 Lectures: 13 Order statistics: definition, Derivation of p. d. f. of i th order statistics, for a random sample of size n from a continuous distribution. Density of smallest and largest observations. Distribution of the sample median when n is odd. Derivation of joint p. d. f. of i th and j th order statistics, statement of distribution of the sample range. Unit-4 Lectures: 10 Discrete and continuous bivariate random variables: Definitions, computation of probabilities of various events, marginal, conditional, product moments and correlations. Conditional expectation and conditional variance. The p. d. f. of a bivariate normal distribution, Marginal and conditional distributions, conditional expectation and conditional variance, regression of Y on X and of X on Y., independence and uncorrelated-ness imply each other, m. g. f and moments. Distribution of ax + by + c, where a, b and c are real numbers. Plotting of bivariate normal density function. 1. Rohatgi V. K. and Saleh A. K. Md. E., An Introduction to probability and Statistics. John Wiley & Sons (Asia). 2. Hogg R.V. and Criag A.T.: Introduction to Mathematical Statistics (Third edition), Macmillan Publishing, New York. 3. Walpole R.E. & Mayer R.H.: Probability & Statistics, MacMillan Publishing Co. Inc, New York 4. Mayer P.L.: Introductory probability & Statistical Applications. Addison Weseley Publication Co., London. 5. Goon A.M., Gupta A.K. and Dasgupta B.: Fundamentals of Statistics (Vol. II) World Press, Calcutta. 6. H. A. David, H. N. Nagaraja, Order Statistics, Third Edition, John Wiley & Sons, Inc 12

13 PAPER CODE BST 302 Sample Survey CREDIT 04 (3-1-0) Unit-1 Lectures: 10 Basic concept: Elementary units, sampling frame, random and non-random sampling. Sampling, census advantages of sampling, Questionnaire and its characteristics. Simple random sampling: Simple random sampling from finite population of size N with replacement (SRSWR) and without replacement (SRSWOR): Definitions, population mean and population total as parameters, inclusion probabilities. Sample mean as an estimator of population mean, derivation of its expectation. Estimation of population proportion: Sample proportion (p) as an estimator of population proportion (P), derivation of its expectation, using SRSWOR. Determination of the sample size. Unit-2 Lectures: 15 Real life situations where stratification can be used, Description of stratified sampling method where sample is drawn from individual stratum using SRSWOR method. Estimator of population mean, population total, derivation of its expectation. Problem of allocation: Proportional allocation, Neyman s allocation and optimum allocation, derivation of the expressions for the standard errors of the above estimators when these allocations are used. Gain in precision due to stratification, comparison with SRSWOR, stratification with proportional allocation and stratification with optimum allocation. Cost and variance analysis in stratified random sampling Unit-3 Lectures: 11 Systematic Sampling: Real life situations where systematic sampling is appropriate, Technique of drawing a sample using systematic sampling, Estimation of population mean and population total, Comparison of systematic sampling with SRSWOR and stratified sampling in the presence of linear trend. Idea of Circular Systematic Sampling. Cluster Sampling: Real life situations where cluster sampling is appropriate, Technique of drawing a sample using cluster sampling, Estimation of population mean and population total (with equal size clusters) Unit-4 Lectures: 9 Ratio Method: Concept of auxiliary variable and its use in estimation, Situations where Ratio method is appropriate, Ratio estimators of the population mean and population total and their standard errors (without derivations), Relative efficiency of ratio estimators with that of SRSWOR. Regression Method: Situations where Regression method is appropriate, Regression estimators of the population mean and population total and their standard errors (without derivations). 1. Cochran, W.G: Sampling Techniques, Wiley Eastern Ltd., New Delhi. 2. Sukhatme, P.V., Sukhatme, B.V. and Ashok A. : Sampling Theory of Surveys with Applications, Indian Society of Agricultural Statistics, New Delhi. 3. Murthy, M.N: Sampling Methods, Indian Statistical Institute, Kolkata. 4. Daroga Singh and Choudhary F.S.; Theory and Analysis of Sample Survey Designs,Wiley Eastern Ltd., New Delhi. 5. Mukhopadhay, Parimal: Theory and Methods of Survey Sampling, Prentice Hall. 13

14 PAPER CODE BST 303 Time Series & Index Numbers CREDIT 04 (3-1-0) Total hours 45 Unit-1 Lectures: 5 Type of data in association with auto correlation, types of dependency, test of independency Unit-2 Lectures: 20 Meaning and utility of index numbers, problems in construction of index numbers. Types of index numbers: price, quantity and value, unweighted and weighted index numbers using (i) aggregate method, (ii) average of price or quantity relative method (A.M. or G.M. is to be used as an average). Index numbers using; Laspeyre s, Paasche s and Fisher s formula. Tests of index numbers: unit test, time reversal test and factor reversal test. Cost of living index number: definition, problems in construction, construction by using (i) Family Budget and (ii) Aggregate expenditure method. Shifting of base, splicing, deflating and purchasing power of money. Uses of index numbers. Unit-3 Lectures: 20 Meaning and need of time series analysis, components of time series, additive and multiplicative model, utility of time series, concepts of autocorrelation and homoscadicityity and heteroscedasticity (without prof), preliminary concept of linear and non-linear time series. 1. Mukhopadhyay, P.:Applied Statistics, new Central Book Agency Pvt. Ltd., Calcutta. 2. Goon A.M., Gupta M.K. and Das Gupta B.: Fundamentals of Statistics, Vol. II, World Press, Calcutta. 3. Chatfield C.: The Analysis of Time Series, IInd Edision Chapman and Hall. 4. Box, G. E. P. and Jenkins, G. M.: Time Series Analysis Forecasting and Control, Holden day, San Francisco. 14

15 PAPER CODE BST 304 Statistical Inference II CREDIT 04 (3-1-0) Unit-1 Lectures: 20 Statement and proof of Cramer Rao inequality. Definition of Minimum Variance Bound Unbiased Estimator (MVBUE) of ϕ (θ), (statement only). Proof of the following results: (i) If MVBUE exists for θ then MVBUE exists for ϕ (θ), if ϕ(.) is a linear function. (ii) If T is MVBUE for θ then T is sufficient for θ. Examples and problems. Definition of MVUE, Procedure to obtain MVUE (statement only), examples. Minimum Variance Unbiased Estimator (MVUE) and Uniformly Minimum Variance Unbiased Estimator (UMVUE), complete sufficient statistic and uniqueness of UMVUE whenever it exists. Unit-2 Lectures: 20 Review of testing of hypothesis and examples of construction of MP test of level α for binomial, Poisson, uniform, exponential and normal models. Testing for one sided and two sided alternatives: Power function of a test, Monotone likelihood ratio properties, definition of uniformly most powerful (UMP) level α test. Statement of the theorem to obtain UMP level α test for one-sided alternative. Illustrative examples. Likelihood Ratio Test (LRT) and its properties: LRT for (i) mean and variance of normal population. (ii) The difference of two means and ratio of two variances of normal populations. Unit-3 Lectures: 05 The need and the concept of confidence interval, Pivotal method of confidence interval, Confidence interval for proportion, mean and variance of normal distribution. Large sample Confidence interval. 1. George Casella, Roger L. Berger (2002), Statistical Inference, 2 nd ed., Thomson Learning. 2. Mukhopadhyay P. (1996): Mathematical Statistics, New central Book Agency (P) Ltd. Calcutta. 3. Rohatgi, V.K. (1984): An Introduction to Probability Theory and Mathematical Statistics, Wiley Eastern. 4. Goon, Gupta & Das Gupta (1991): An Outline of Statistical Theory, Vol. II, World Press. 5. Hogg, R.V. and Craig, A.T. (1971): Introduction to Mathematical Statistics, McMillan. 15

16 PAPER CODE BST 305 Numerical Analysis CREDIT 04 (3-1-0) Unit-1 Lectures: 10 Finite difference theory, basic property, forward and backword difference operators, Difference table(forward and backword), Displacement operator, Properties of operators, and E, n th difference of (x), Difference of some special functions such as Trigonometrical, Exponential, Logarithmic, P n Relation between E and and E and, Method of separation of symbols, Factorial polynomial. Difference of a factorial polynomial. Unit-2 Lectures: 10 Interpolation with equal interval: Interpolation meaning, assumptions and accuracy, method of interpolation, Newton s Gregory forward and backword interpolation formulae for equal intervals, Newton s advancing difference formula. Applications of Newton s formula, Estimate of missing value of f(x) with the help of known values. Binomial expansion method (only one or two missing values). Unit-3 Lectures: 12 Interpolation with unequal intervals: Divided differences with divided difference table, Properties of divided differences, Newton s divided difference formula, Relation between divided and ordinary differences. Lagrange s Interpolation formula. Central difference formulae due to Gauss (Forward and backword),sterling s formula, Bessel s formula. Unit-4 Lectures: 13 Numerical Integration: General Quadrature formula, Trapezoidal rule, Simpson s one third and three eight rules, Weddle s rule, Eular Maclaurin s summation formula and its application. Solution of algebraic and transcendental equations synthetic division method, graphical method Regula falsi method, Newton s Raphson method. Solution of ordinary difference equations (first order), Eular s methods Eular s modified method, Picard s methods. 1. Operation Research by Kantiswaroop, P. K. Gupta & Man Mohan. 2. Numerical methods: problems and solutions by Jain, Iyengar and Jain, New Age International Publisher. 3. An introduction to numerical analysis (2 nd Edition) by K.E. Atkinson, John Wiley, Numerical methods and analysis by J.I. Buchanan and P.R. Turner, McGraw-Hill, Numerical analysis, Bansal and Ojha, Jaipur publishing house Jaipur. 16

17 PAPER CODE BST 306 Practicals CREDIT 04 (0-0-4) CONTENT Students will be required to do practicals, based on topics listed below, using MS Excel, : 1. Model sampling from Gamma and log normal distributions. 2. Sketching of p.m.f., d.f. of Negative Binomial and Hypergeometric distributions 3. Bivariate Discrete distribution I (Marginal & conditional distribution, computation of probabilities of events. Expectations/conditional expectations/ variances / conditional variance/covariance /correlation coefficient) 4. Model sampling from bivariate normal distribution and Application of bivariate normal distribution. 5. Simple random sampling with and without sampling 6. Stratified random sampling. 7. Systematic Sampling. 8. Ratio Method of Estimation. 9. Regression Method of Estimation. 10. Determination of secular trend by moving averages and least squares methods. 11. Construction of index numbers. 12. Tests for consistency of index numbers. 13. Construction of Consumer Price Index - interpretation. 14. Interval estimation of location and scale parameters of normal distribution and population proportion Interval estimation for difference between two population proportions. 15. Likelihood ratio test 16. General quadrature formulae and Missing values 17. Newton s Gregory forward and backword interpolation (equal interpolation) 17

18 PAPER CODE BST 351 Elements of Reliability & Survival Analysis CREDIT 04 (3-1-0) TOTAL HOURS 44 Unit-1 Lectures: 10 Preliminaries: Definition and concept of time, event, Survival function, quantiles, hazard rate, cumulative hazard function and their relation with survival function mean residual life. Parametric models: Exponential, Weibull and normal and their survival characteristics. Unit-2 Lectures: 12 Censoring mechanisms- type I, type II and left right and interval censoring. Likelihood function under censoring, Fitting parametric models to survival data without censoring and with right censoring. Unit-3 Lectures: 12 System and its Configuration, Series Configuration, Parallel Configuration, Series -Parallel Configuration, Parallel - Series Configuration. Unit-4 Lectures: 10 Empirical survival function, Actuarial estimator, Kaplan Meier estimators and its properties. 1. Deshpande, J.V. and Purohit, S. G.(2005): Life Time Data: Statistical Model and Methods, World Scientific. 2. Cox, D. R. and Oakes, D. (1984): Analysis of Survival Data, Chapman and Hall, New York. 3. Sinha, S. K. and Kale, B. K. (1983): Life Testing and Reliability Estimation, Wiley Eastern Limited. 4. Elandt Johnson, R.E. Johnson N. L.: Survival Models and Data Analysis, John Wiley and Sons. 5. Miller, R. G. (1981): Survival Analysis (John Wiley) 18

19 PAPER CODE BST 352 Design of Experiments CREDIT 04 (3-1-0) Unit-1 Lectures: 15 Introduction, One factor analysis, the model and problem Decomposition of total sum of squares, independence of decomposed sum of squares, ANOVA table, two factor analysis ANOVA table, Test for normality. Basic terms in design of experiments: Experimental unit, treatment, layout of an experiment. Basic principles of design of experiments: Replication, randomization and local control. Choice of size and shape of a plot for uniformity trials, the empirical formula for the variance per unit area of plots. Unit-2 Lectures: 15 Complete randomized design, randomized block design and Latin square design. Layout, model, assumptions and interpretations: Estimation of parameters, expected values of mean sum of squares, components of variance. Tests and their interpretations, test for equality of two specified treatment effects, comparison of treatment effects using critical difference (C.D.). Efficiency of design: Concept and definition of efficiency of a design. Efficiency of RBD over CRD. Efficiency of LSD over CRD and LSD over RBD. Identification of real life situations where CRD, RBD and LSD are used. Unit-3 Lectures: 15 General description of factorial experiments, 2 2 and 2 3 factorial experiments arranged in RBD. Definitions of main effects and interaction effects in 2 2 and 2 3 factorial experiments. Model, assumptions and its interpretation. General idea and purpose of confounding in factorial experiments. Total confounding (Confounding only one interaction): ANOVA table, testing main effects and interaction effects. Construction of layout in total confounding, confounding in 2 3 factorial experiments. 1. Goon, Gupta, Dasgupta : Fundamental of Statistics, Vol. I and II, The World Press Pvt. Ltd. Kolkata. 2. Montgomery, D.C.: Design and Analysis of Experiments, Wiley Eastern Ltd., New Delhi. 3. Cochran, W.G. and Cox, G.M.: Experimental Design, John Wiley and Sons, Inc., New York. 4. Gupta, S.C. and Kapoor, V.K. : Fundamentals of Applied Statistics, S. Chand & Sons, New Delhi. 5. Das, M.N. and Giri, N.C. : Design and Analysis of Experiments, Wiley Eastern Ltd., New Delhi. 19

20 PAPER CODE BST 353 Statistical Process Control & Vital Statistics CREDIT 04 (3-1-0) Unit-1 Lectures: 10 Meaning and purpose of Statistical Process Control, Concept of Quality and Quality Control, process control, product control, assignable causes, chance causes and rational subgroups Unit-2 Lectures: 15 Control charts and their uses, Choice of subgroup sizes, Construction of control chart for X (mean), R (range), s (standard deviation), c (no. of defectives), p (fraction defectives) with unequal subgroup size. Interpretation of non-random patterns of points. Modified control chart. Unit-3 Lectures: 10 Census, Registrar, Ad-hoc surveys, Hospital records, Demographic profiles of the Indian census. Crude death rate, Age-specific death rate, Infant mortality rate, Death rate by cause, standardized death rate. Central Mortality Rate, Force of Mortality Rate. Unit-4 Lectures: 10 Description and construction of complete and abridged life tables and their uses. Construction of complete life table from population and death statistics. Stable and Stationary populations. 1. Mukhopadhyay, P. (1994): Applied Statistics, new Central Book Agency Pvt. Ltd., Calcutta. 2. Goon A.M., Gupta M.K. and Das Gupta B. (1986): Fundamentals of Statistics, Vol. II, World Press, Calcutta. 3. Srivastava, O.S. (1983) : A text book of demography. Vikas Publishing House, New Delhi. 4. Duncan A.J. (1974) : Quality Control and Industrial Statistics, IV Edision, Taraporewala and Sons. 5. Benjamin, B. (1959) : Health and vital statistics. Allen and Unwin 20

21 PAPER CODE BST 354 Operation Research CREDIT 04 (3-1-0) Unit-1 Lectures: 15 Nonlinear Programming: Unconstrained algorithms; direct search method, gradient method. Constrained methods; Separable programming, quadratic programming. General Inventory models, role of demand in the development of inventory; Static Economic-Order- Quantity (EOQ) models; Dynamic EOQ models. Continuous review model, single period models, multi period models. Unit-2 Lectures: 15 Elements of Queuing models, role of exponential, pure birth and death models. Generalized Poisson Queuing models, Specialized Poisson Queues: Steady state measures of performance, single server model multi server models, machine servicing models-(m/m/r): (GD/K/K), R< K. Replacement and maintenance models; gradual failure, sudden failure, replacement due to efficiency deteriorate with time, staffing problems, equipment renewal problems. Unit-3 Lectures: 15 Simulation modeling: Monte Carle Simulation, Types of simulations, Elements of discrete-events simulation, generation of random numbers. Mechanics of discrete simulation, Methods of gathering statistical observations: subinterval method, replication method, regeneration method. Sequencing Problems: notions, terminology, and assumptions, processing n jobs through m machines. 1. Operations Research an Introduction Hamady A. Taha, Prentice Hall. 2. Operations Research Theory and Applications-J. K. Sharma, Macmillan Publishers. 3. Non linear Programming S.D. Sharma, Kedar Nath Ram Nath & Co. 4. Mathematical Programming Theory and Methods-S.M.Sinha. 5. Operations Research, - Kanti Swarup, P. K. Gupta and Man Mohan, Sultan Chand & Sons. 21

22 PAPER CODE BST 355 Official Statistics & Demand Analysis CREDIT 04 (3-1-0) Unit-1 Lectures: 8 Present official statistical system in India relating to census and population; methods of collection of official statistics. Various agencies responsible for the data collection: Central Statistical Organization (CSO), the National Sample Survey Organization (NSSO), office of Registrar General, their main functions and important publications. State Government organizations. Unit-2 Lectures: 7 Agricultural Statistics: Area and Yield statistics. National Income statistics: Income, expenditure and production approaches. Their applications in various sectors in India. Unit-3 Lectures: 15 Demand Analysis: Introduction, Necessities and Luxuries, Demand and Supply, Laws of Demand and Supply: Demand and Supply Curve. Price Elasticity of Demand: Significance of Elasticity of Demand, Demand Function with constant price elasticity. Price Elasticity of Supply: Supply Curve with the constant Price Elasticity. Partial Elasticities of Demand. Unit-3 Lectures: 15 Types of Data Required for Estimating Elasticities: Family Budget Data (cross section data) and Market Statistics or Time Series Data. Methods of Estimating Demand Function: Leontief s Method and Pigous s Method. Engel s Law and Engel s Curve. Pareto s Law of Income Distribution. Utility Function. 1. Gupta S. C. and Kapoor V. K. (2008): Fundamentals of Applied Statistics. 4 th Edn. (Reprint) Sultan Chand & Sons, New Delhi. 2. C. S. O. (1984): Statistical System in India. 3. Goon A. M.,Gupta M. K,, and Dasgupta. B. (2005): Fundamentals of Statistics. Vol.-II, 8 th Edn. World Press, Kolkata. 4. Mukhopadhyay P. (1999): Applied Statistics. Books and Allied (P) Ltd. 5. Nagar A. L. & Das R. K. (1976): Basic Statistics 6. Elhance, D. N. and Elhance, V. (1996): Fundamentals of Statistics. D.K. Publishers. 7. M.R.Saluja : Indian Official Statistics. ISI publications. 22

23 PAPER CODE BST 356 Practicals CREDIT 04 (3-1-0) CONTENT Students will be required to do practicals, based on topics listed below, using MS Excel, : 1. Analysis of CRD. 2. Analysis of 2 2 factorial experiment using RBD layout. 3. Analysis of 2 3 factorial experiment using RBD layout. 4. Analysis of 2 3 factorial experiment using RBD layout. (Complete confounding) 5. Markov Chains: realizations, classification of states. 6. Poisson process: realizations, examples 7. Computation of various mortality and fertility rates. 8. Construction of life table and computation of expectation of life and force of mortality. 9. Plotting of survival function, hazard rate for probability distributions. 10. Kaplan-Meier Estimator. 11. Demand and Supply Curve 12. Analysis of demographic data (India) 13. Analysis of agricultural data (India) 14. X R charts. (Standard values known and unknown) 15. np and p charts. (Standard values known and unknown). 16. Laws of Income Distributions 17. Linear programming (graphical methods) 18. Simplex method 19. Transportation problems. 23

24 PAPER CODE MST 101 Probability Theory Unit-1 Lectures: 10 Random experiment, outcomes, sample space, the class of subsets (events), Borelfield(σ-field), axiomatic definition of probability measure, combination of events. Bonferroni and Boole inequalities. Independence of events (pairwise and complete independence).sequences of events.borel-cantelli lemma. Conditional probability, laws of total and compound probability, Bayes Theorem. Unit-2 Lectures: 15 Concept of a random variable on a finite probability space and its probability distribution.probability mass function and cumulative distribution function (cdf) of a random variable.extension of the concept to any discrete random var-iable and an absolutely continuous random variable. Unit-3 Lectures: 10 Expectation of a random variable, proper-ties of expectation, conditional expectation and its properties. Moments, quantiles, and variance.properties of a cdf. Bivariate distributions and the joint probability distribution. Independence of random variables. Marginal and conditional distributions. Moment generating function, probability generating function, characteristic function and their properties.inversion, continuity and uniqueness theorems. The moment problem.demoivre-laplace Central Limit Theorem, Central Limit Theorem for IID random variables, Central Limit Theorem. Unit-4 Lectures: 10 Sequences of random variables.convergence in distribution and in probability. Almost sure convergence and convergence in the r th mean. Implication between modes of convergence.slutsky s theorem. Tchebychev and Khintchine weak laws of large numbers. Kolmogorov inequality(statement). Kolmogorov strong law of large numbers (statement). 1. Bhat, B.R. (1999). Modern Probability Theory, 2/e, New Age International, New Delhi. 2. Rao B. L. S. Prakasa (2009) A First course in Probability and Statistics. World Scientific 3. Meyer, P.A. An Introduction to Probability and Its Applications. PHI 4. John E Freund (2004): Mathematical Statistics with applications. 7 th ed. Upper saddle River, NJ: Prentice Hall. ISBN: Rohatgi V.K & A.K. MD. Ehsanes Saleh (2001): An Introduction to Probability Theory andmathematical Statistics, 2 nd. John Wiley and Sons. 24

25 PAPER CODE MST 102 Distribution Theory Unit-1 Lectures: 8 Discrete Distributions: Binomial, Poisson, multinomial, hypergeometric, negative binomial, uniform. The (a,b,0) class of distributions. Moments, quantiles, cdf, survival function and other properties. Unit-2 Lectures: 15 Continuous Distributions: Uniform, Normal, Exponential, gamma, Weibull, Pareto, lognormal, Laplace, Cauchy, Logistic distributions; properties and applications. Functions of random variables and their distributions using Jacobian of transformation and other tools. Bivariate normal and bivariate exponential distributions. Unit-3 Lectures: 12 Concept of a sampling distribution. Sampling distributions of t, χ 2 and F (central and non central), their properties and applications. Multivariate normal distribution. Distribution of a linear function of normal random variables. Characteristic function of the multivariate normal distribution. Distribution of quadratic forms. Cochran s theorem. Independence of quadratic forms. Unit-4 Lectures: 10 Compound, truncated and mixture distributions. Convolutions of two distributions. Order statistics: their distributions and properties. Joint, marginal and conditional distribution of order statistics. The distribution of range and median. Extreme values and their asymptotic distribution (statement only) with applications. 1. Rohatgi V.K & A.K. MD. Ehsanes Saleh: An Introduction to Probability Theory and Mathematical Statistics, 2 nd. John Wiley and Sons, Johnson, Kotz and Balakrishna, Continuous univariate distributions, Vol- 1 IInd Ed, John Wiley and Sons 3. Johnson, Kemp and Kotz, Univariate discrete distributions, IIInd Ed, John Wiley and Sons 4. Mukhopadhyay P. (1996): Mathematical Statistics, New central Book Agency (P) Ltd. Calcutta. 5. Goon, Gupta & Das Gupta (1991): An Outline of Statistical Theory, Vol. I, World Press. 25

26 PAPER CODE MST 103 Linear Algebra and Matrix Theory Unit-1 Lectures: 12 Linear equations and matrices, matrix operations, solving system of linear equations, Gauss-Jordan method, Concept & Computation of determinant and inverse of matrix, Eigen values and eigen vectors, illustrations of the methods, Positive semi definite and position definite matrices, illustrations. Unit-2 Lectures: 12 Overview. Fields (real and complex numbers are special cases), and the notion of a vector space over a field. Subspaces. Sums and direct sums of subspaces. Criteria for a sum to be a direct sum. Examples. Linear combinations and spans. Finite dimensional vector spaces. Linear independence and the notion of basis. More on basis and dimension. Unit-3 Lectures: 13 More about the structure of linear maps on complex vector spaces: Jordan normal form. The minimal polynomial and relation to characteristic polynomial. Analogous (but weaker) results for linear maps on real vector spaces: existence of 1 or 2 dimensional invariant subspaces block upper triangular matrices, the characteristic polynomial of 2x2 matrices. Eigen pairs and generalized Eigen pair-spaces. Decomposition into generalized Eigen spaces and Eigen-pair spaces. Unit-4 Lectures: 8 More properties of the adjoint of a linear map between inner product spaces (behavior with respect to composition, sums and scalar products, kernels and images). Self-adjoint operators: first properties. Normal operators (a larger class than self-adjoint), and their characterization. The complex and real spectral theorems. 1. Searle, S. R. (1982). Matrix Algebra Useful for Statistics; John Wiley, New York. 2. Ramachandra Rao, A. and Bhimasankaram, P. (1992): Linear Algebra, Tata McGrawhill. 3. Krishnamurthy V., Mainra V.P. and Arora J. L. (2009) An introduction to Linear Algebra, East-West Press Pvt Ltd. 4. Rudin, W. (1985). Principles of Mathematical Analysis, McGrawhill, New York. 5. Malik, S.C. and Arora, S. (1998). Mathematical Analysis, New Age, New Delhi. 26

27 PAPER CODE MST 104 Sampling Theory Unit-1 Lectures: 12 Concept of population and sample, Need for Sampling, census & sample surveys, basic concepts in sampling and designing of large-scale surveys design, sampling scheme and sampling strategy. Basic methods of sample selection: SRSWR, SRSWOR. Stratification, Allocation and estimation problems. Construction of Strata: deep stratification, method of collapsed strata. Unit-2 Lectures: 17 Systematic sampling: The sample mean and its variance, comparison of systematic with random sampling, comparison of systematic sampling with stratified sampling, comparison of systematic with simple and stratified random sampling for certain specified population. Estimation of variance, Two stage sample: Equal first stage units, Two stage sample: Unequal first stage units; systematic sampling of second stage units. Unequal probability sampling: PPSWR/WOR methods (including Lahiri s scheme) and Des Raj estimator, Murthy estimator (n=2). Horvitz Thompson Estimator of a finite population total/mean, Midzuno sampling. Unit-3 Lectures: 11 Cluster sampling. Two stage sampling with equal number of second stage units. Use of supplementary information for estimation: ratio and regression estimators and their properties. Unbiased and almost unbiased ratio type estimators, Double sampling. Unit-4 Lectures: 05 Non-sampling error with special reference to non-response problems. Hansen Horwitz and Demig s techniques. 1. Sukhatme P. V., Sukhatme B. V.& Ashok C : Sampling Theory of surveys with applications Indian Society of Agricultural Statistics, New Delhi. 2. Des Raj and Chandhok. P. (1998) : Sample Survey Theory - Narosa publication. 3. William G. Cochran. ( 1977) : Sampling Techniques- IIIrdedition John and Wieley sons Inc. 4. Parimal Mukhopadhyay (1998) : Theory and methods of survey sampling Prentice Hall of India private limited. 5. Murthy M.N. (1977) : Sampling Theory of Methods - Statistical Publishing Society, Calcutta. 27

28 PAPER CODE MST 105 R: Statistical Programming Language Unit-1 Lectures: 12 Introduction, installation, start and quite, help.search(), console and script, Saving workspace/history. Advantages, packages. Data handling in R: numeric/character/logical type data, real/integer/complex type data, matrices with their operations, data frames, lists, creating arrays/vectors. Arithmetic and logical operators, some mathematical expressions such as logarithm, square-root, exponentiation etc. data editor. Unit-2 Lectures: 10 Control structures: conditional statements - if and if else; loops - for, while, do-while; nested loops, functions built-in and user defined, Import and export of excel and text sheets. Unit-3 Lectures: 10 High level, Low level and Interactive graphics, plot command, histogram, barplot, boxplot, Pie charts, inserting mathematical symbols in a plot, Customisation of plotsetting graphical parameters, adding text, saving to a file; Adding a legend. Unit-4 Lectures: 13 Descriptive statistics; mean, variance, correlation and summary statistics. Linear models: functions for ANOVA/regression. Test of significance; t tests, chi-square test, Nonparametric tests. Probability Distributions: Obtaining density, cumulative density and quantile values for discrete and continuous distributions; simulations from discrete and continuous distributions, Plotting density and cumulative density curves. 1. Peter Dalgaard, Introductory Statistics with R, Springer, 2nd edition, Michael J. Crawley, The R Book, John Wiley and Sons, Ltd., Richard A. Becker, John M. Chambers and Allan R. Wilks (1988), The New S Language. Chapman & Hall, New York. 4. John M. Chambers and Trevor J. Hastie eds. (1992), Statistical Models in S. Chapman & Hall, New York. 5. John M. Chambers (1998) Programming with Data. Springer, New York. 28

29 PAPER CODE MST 106 Practicals CREDIT 04 (0-0-4) CONTENT Students will be required to do practicals, based on topics listed below, using R software: 1. Convergence of the random variable 2. Fitting of discrete and continuous distributions 3. Sketching of p.m.f./ pdf of discrete/ continuous distributions 4. R- program (User defined) for Matrix operations (Multiplication, determinate, inverse, Eigen values and vector) 5. Simple random sampling with and without sampling 6. Stratified random sampling. 7. Systematic Sampling. 8. Ratio Method of Estimation. 9. Regression Method of Estimation. 10. Unequal probability sampling: PPSWR/WOR methods (including Lahiri s scheme) 11. Cluster sampling There shall be minimum two practical assignments from each optional course. 29

30 PAPER CODE MST 201 Estimation and Testing of Hypotheses Unit-1 Lectures: 13 Concept of mean squared error, Criteria of a good estimator: unbiasedness, consistency, efficiency and sufficiency. Fisher-Neyman factorization theorem, Family of distributions admitting sufficient Statistic. Point estimation, Maximum likelihood method (MLE), moments, Least squares method. Method of minimum chi-square and percentiles. Properties of maximum likelihood estimator (with proof). Successive approximation to MLE, Method of scoring and Newton-Raphson method. Unit-2 Lectures: 10 Cramer-Rao inequality and its attainment, Cramer-Huzurbazar theorem (statement only), Completeness and minimal sufficient statistic, Ancillary statistic, Basu theorem, Uniformly minimum variance unbiased estimator (UMVUE). Rao-Blackwell and Lehmann-Scheffe theorems and their applications, Review of convergences of random variables and their implications, Delta method and its application, Asymptotic efficiency and asymptotic estimator, consistent asymptotic normal (CAN) estimator. Unit-3 Lectures: 12 Statistical Hypothesis, critical region, types of errors, level of significance, power of a test, Test function, Randomized and non-randomized tests, Most powerful test and Neyman-Pearson lemma. MLR family of distributions, unbiased test. Uniformly most powerful test. Uniformly most powerful unbiased test. Likelihood ratio test with its properties. Unit-4 Lectures: 10 Confidence interval, confidence level, construction of confidence intervals using pivots, Determination of confidence intervals based on large and small samples, uniformly most accurate one sided confidence interval and its relation to UMP test for one sided null against one sided alternative hypotheses. 1. George Casella, Roger L. Berger, Statistical Inference, 2nd ed., Thomson Learning. 2. Mukhopadhyay P.: Mathematical Statistics, New central Book Agency (P) Ltd. Calcutta. 3. Rao, C.R.: Linear Statistical Inference and its Applications, 2nd ed, Wiley Eastern. 4. Rohatgi, V.K.: An Introduction to Probability Theory and Mathematical Statistics, Wiley Eastern. 5. Mood, Graybill and Boes.: Introduction to the theory of Statistics 3rd ed., McGraw- Hill. 6. Goon, Gupta & Das Gupta: An Outline of Statistical Theory, Vol. II, World Press. 7. Hogg, R.V. and Craig, A.T.: Introduction to Mathematical Statistics, McMillan. 8. Kendall, M.G. and Stuart, A. : An Advanced Theory of Statistics, Vol. I,II. Charles Griffin. 9. Kale, B.K. : A First Course on Parametric Inference, Narosa Publishing House. 10. Lehmann, E.L.: Testing Statistical Hypotheses, Student Editions. 30