**1. General Statistics**

a) Measures of central tendencies and dispersions, Skewness and kurtosis. Association of attributes, simple, multiple and partial correlation and regressions.

b) Various Index numbers and test for Index numbers.

c) Life table, birth and death ratio, measures of fertility and reproduction. Various indices of measuring population growth, Gross and Net reproduction rates.

d) Control charts for variables and attributes, Single sampling plans for attributes, OC function, ASN, AQL, LTPD and AOQL, Producers and consumers risk.

e) Reliability function, mean time to failure, failure and hazard rates for normal, exponential, Weibull, log normal and gamma failure laws. Reliability of series and parallel systems.

2. Probability and probability distribution :

a) Definition of probability (classical, mathematical and axiomatic). Addition and multiplication theorems of probability, conditional probability and independent events. Bayes theorem.

b) Probability density function / probability mass function and cumulative distribution function. Relation between central and raw moments. Moment generating function and probability generating function.

c) Simple properties of (i) Discrete distributions; Uniform, Bernoulli, bionomial, Poisson, geometric, negative binomial, hyper geometric (ii) Continuous distribution : Uniform, exponential, gamma and normal.

d) Continuous and discrete bivariate distributions, their marginal and conditional distributions, Conditional mean and variance. Bivariate normal distribution.

e) Chebyshev’s inequality. Distribution of functions of one and two random variables. Order statistics.

3. Statistical Inference :

a) Unbiased, consistent, sufficient and efficient estimators, C-R inequality, Rao-Blackwell, and Lehmann – Scheffe theorems and their use in obtaining minimum variance unbiased estimators.

b) Method of estimation moments, maximum likelihood and minimum c2 methods.

c) Null and alternative hypothesis, simple and composite hypotheses, critical region, error of type I and type II. Size and power of a test Most Powerful (MP) and uniformly most powerful (UMP) tests, Neyman Pearson lemma, LRT, their application in construction of tests.

d) Test based on normal, t, c2 and F-distributions.

e) Sequantial probability ratio tests (SPRT) for simple versus simple hypothesis, Relations between a, b, A and B, Walds fundamentals identity, OC and ASN functions, Simple examples based on normal, binomial, Poisson and exponential distributions.

4. Sampling, Design of experiments :

a) Sampling and non sampling errors, Simple random sampling with and without replacement, Estimates of population mean, total and proportion, Variances of these estimates and estimates of these variances.

b) Stratified random sampling, estimation of population and total. Variances of these estimates, Proportional and optimum allocations and their comparison with SRS.

c) Systematic sampling, estimates of population mean and total, variance of these estimates.

d) Ratio and regression methods of estimation, estimation of population mean and total, variances of estimates and estimates of these variances, variances in terms of population correlation coefficient between X and Y for regression method and their comparison with SRS.

e) Basic priciples of experimental design. Layout and analysis of completely randomized design (CRD(), Randomized Black Design (RBD) and LSD, Estimation and analysis of one missing observation.

f) Analysis of 22 and 23 factorial experiments.

5. Linear Programming, Games Theory and Operations Research :

a) Basic concept : Linear independent vectors,basic, matrix and its inverse, solution of simultaneous linear equations. Convex sets, extreme points of a convex set.

b) Feasible and basic feasible solution to an LPP and their properties.

c) Theory of simplex method : computational procedure for solving an LPP problem. Artificial basis technique.

d) Dual linear programming : duality theorem; dual simplex algorithm.

e) Two person zero sum games; the Minimax principle; Saddle point, Pure and mixed strategies, Graphical solution of 2 x 2 and mx2 games; Dominance principle; reducing a game o an LPP.

f) Transportation and Assignment Problem : Transportion and Assignment problems, Methods of selection of an intial basic feasible solution to a TP, Solution of TP by UV method, Resolution of degeneracy in TP, Hungarian method for solving an Assignment problem.

g) Inventory Problems : Components of an inventory system; Deterministic EOQ models with discrete and continuous demands with and without shortage costs. A single period stochastic model with no setup cost having zero or non zero intial stock.

h) Replacement Problem : Replacement of items that deteriorate with time and for which the maintenance cost increases with time; Replacement of items that fail completely; Individual and group replacement policies; System Reliablity.

i) Sequencing Problems : The sequencing problems with n-jobs and two machines; Optimal sequencing algorithm; problems with n-jobs and three machines.

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