## BASIC STATISTICS

Definition of statistics Scope and limitations of statistics, Primary and Secondary data collection and presentation of data,summarizing data,frequency distribution,Measures of location , mean median mode (Simple problems) Percentiles.

Measures of variability Range,Quartile deviation,Standard deviation,coefficient of variation,Moments about the origin and mean, Skewness, Kurtosis and their measures.

Measures of association between attributes coefficient of association and contingency; Measures of relation between two variables,correlation and regression,Curve fitting by least squares.

Partial and Multiple correlation coefficients (three variables only), Rank correlation,Simple problems.

Present official statistical system in India relating to population, agriculture, Industrial production, trade and prices; Methods of collecting official statistics, their reliability and limitations; Principal publications containing such statistics; Official agencies responsible for their collection and their functions.

## BASIC PROBABILITY THEORY

Events, Sample Space, Mathematical and Statistical definitions of Probability, Axiomatic definition of Probability,Addition & multiplication theorems,conditional probability,Bayes Theorem,Simple problems.

Random variable, Discrete and Continuous Random variables,Distribution function and its properties,Expectation,Moment Generating function,Probability Generating function.Chebyshev s inequality,Cauchy,Schwartz inequality,characteristic function, its properties and uses.

Concept of Bivariate distributions - conditional and marginal distributions -Notion of Independence of Random variables - Conditional Expectation - Simple problems. Weak Law of Large numbers (WLLN) - Bernoulli's Theorem, convergence in probability and distributions - Central Limit theorem for iid case. PROBABILITY DISTRIBUTIONS

Discrete Distributions: Bernoulli, Binomial - Poisson - Geometric - Multinomial distributions and their characteristics.

Continuous Distributions: Uniform - exponential - Normal - Gamma - Beta distributions and their characteristics.

Cauchy distribution - Laplace distribution - Bivariate Normal distribution - Conditional and marginal distributions

Sampling distributions - standard error - Sampling distributions t, F and chi-square distributions - Interrelationship among t, F and chi-square distributions and their characteristics

Concept of Order statistics - Distribution of the Order Statistics including that of maximum and minimum - Distribution of the sample Range and median

## SAMPLING THEORY

Concept of sampling - Need for sampling - population and sample - sampling unit and sample frame - Types of Population - Basic properties of population - sample survey and census - Principal steps in a Sample survey - Notion of sampling error.

Simple Random Sampling with and without replacement - Estimation of Population mean and proportion and their variances- Determination of sample size.Stratified sampling - Principles of stratification - Estimation of population mean and its variance - Allocation techniques - Estimation of gain due to stratification

Systematic sampling - Estimation of population mean and its sampling variance - Circular systematic sampling - comparison of systematic, simple random and stratified random sampling - cluster sampling with equal sized clusters - estimation of population mean and variance.

Large scale sample surveys - Sources of Non sampling errors and methods of controlling them - NSS and CSO and their functions.

## BASIC ESTIMATION THEORY

Basic problem of statistical Inference; Point estimation; Properties of estimators; Unbiasedness and consistency; conditions for consistency; sufficiency; factorisation theorem (without proof) - Applications.

Efficiency; minimum variance unbiased estimators and their properties ; Cramer-Rao Inequality, Rao - Blackwell Theorem and their applications.

Methods of Estimation: Methods of moments, least square and minimum chi-square methods; Statement of their properties and applications.

Method of maximum likelihood and its applications; properties of maximum likelihood estimators, asymptotic properties (without proof). Bayes Estimators: Notions of Prior and Posterior distributions, improper and conjugate prior and Bayes’ Estimators.

Confidence intervals: Basic Notions; Confidence Intervals for the mean, proportion and variance (for the case of one and two populations) and correlation coefficients- Large sample Confidence Intervals

## ELEMENTS OF TESTING STATISTICAL HYPOTHESES

Neyman -Pearson formulation of the Hypothesis testing problem; concept of hypotheses - Types of errors and power - most powerful tests - Neyman-Pearson Fundamental Lemma and its applications - Notion of Uniformly most powerful tests.

Likelihood Ratio tests: Description and property of LR tests - Application to standard distributions - Large sample properties.Standard tests of significance relating to mean, proportion and variance (for one and two populations), tests for correlation coefficients - Large sample tests

Non Parametric Tests: Sign test, Signed rank test, Median test , Mann-Whitney test, Goodness of fit test; Chi-square and Kolmogorov Smirnov test (Description, properties and applications only)

Linear Models: Estimation of parameters - Gauss Markov theorem - Tests of significance for the parameters in the model STATISTICAL QUALITY CONTROL AND OPERATIONS RESEARCH

Need for SQC in industries; Process control: Chance and assignable causes of variation; specification and tolerance limits; process capability; statistical basis for control charts, X -R, p and c charts, their construction and analysis.

Product control: Acceptance sampling by attributes; Producer s and Consumer s risk; Notions of AQL, LTPD and AOQL; Single and double sampling plans; OC, AOQ, ASN and ATI Curves.

Linear Programming Problem; Graphical solution; Simplex method; Artificial variables Techniques - M-method and Two-Phase method.

Transportation Problem: North west corner, least cost and Vogel s approximation methods; Assignment problem and its algorithm; Traveling Salesman Problem.
Network analysis by CPM / PERT; Basic concepts: Constructions of the network; concepts of slack and float in network analysis; Determination of the floats and critical path.

## PRINCIPLES OF EXPERIMENTAL DESIGN

Basic Principles for designing statistical experiments: Randomisation, Replication and local control techniques; determination of experimental units and notion of experimental error. Analysis of variance with one way and two way classifications; Models and Methods of analysis.
Completely randomized and randomized block designs - Models and estimates of parameters and their standard error - Analysis of data arising from such designs, Analysis when one or two observations are missing.

Latin Square Design Model Estimation of parameters, Method of analysis, Missing Plot technique in LSD Analysis of covariance, One-way classification only
Multiple Comparison tests: LSD , Student 0Newman Keuls test , Duncan s Multiple range test, Tukey s test - Transformations to stabilize the variance .

Factorial Experiments: 22, 23 and 32 designs; estimation of main effects and interactions and their standard errors - Principles of confounding

## APPLIED STATISTICS

Index Numbers; Construction of index numbers; fixed and chain base index numbers; weighted index numbers; standard index numbers ; Tests for index numbers ; cost of living index number and its construction.

Time Series Analysis: Time Series models - Components of a time series - Methods of trend and isolation,Moving average, Seasonal indices, Ratio to trend, Link relative methods - Cyclical fluctuations

Sources of Demographic data: Measures of mortality, Crude and specific rates, standardized rates, Infant mortality rate, Complete life table , its construction and uses. Abridged life tables

Measures of fertility: CBR, ASBR, GFR and TFR, Crude, Specific and standardized rates - Measures of migration, Population growth rates; GRR and NRR.

Educational Statistics, methods of standardization of scales and tests; Z- scores, standard scores, T- scores and percentile scores; validity of test score and its determination; Intelligence Quotient, its measurement and uses.

Joint and marginal distributions, Characteristic functions, Moment generating functions, Laplace transformation. Conditional distribution and conditional expectation.
Multivariate normal and multinomial random vectors/distributions.
Different form of convergence of random vectors. Chebyshev s inequality, Law of Lager Numbers, Central Limit Theorem, Borel-Cantelli Lemma.

## MARKOV CHAINS

Transition probability matrices. Examples. Classification of states. Periodicity and recurrence.

## LIMIT THEOREMS FOR MARKOV CHAINS AND APPLICATIONS

Stationary distributions. Absorption probabilities.
Conditions for recurrence. Applications to queueing models.

## CONTINUOUS TIME MARKOV CHAINS

Poisson processes. Birth and death processes. Kolmogorov equations.

## RENEWAL PROCESSES

Definitions and examples. Renewal equations and the elementary renewal theorem

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