DIFFERENT ASPECTS OF DATA, AND ITS COLLECTION

Statistics as collected facts and figures, and as a science for extracting information from data. Concepts of a statistical population and sample. Different types of characteristics and data qualitative and quantitative, cross sectional and time-series, discrete and continuous, frequency and non-frequency. Different types of scale- nominal and ordinal, ratio and interval. Collection of data-census and sampling. Different types of random samples- simple random sample, systematic, stratified and cluster sampling.

CENTRAL TENDENCY AND DISPERSION

Averages- Arithmetic Mean, Median, Mode, Geometric Mean, Harmonic Mean and Weighted averages.  Absolute Measures of dispersion- Range, Quartile Deviation, Mean Deviation and Standard Deviation. Combined mean and standard deviation, C.V, relative measures of dispersion, Ogives and Box plot.

Moments, Skewness and Kurtosis

Raw moments, central moments and their inter relations. Skewness- Pearson’s, Bowly’s and moment measures of skewness. Kurtosis- percentile and moment measure of kurtosis.

INDEX NUMBERS

Definition of Index Numbers. Price Index Numbers. Price Index Numbers as Simple (A. M., G. M.) and Weighted averages (A. M.) of price relatives. Weighted averages (A. M.) of price. Laspeyer’s, Paasche’s and Fisher’s Index Numbers. Time-Reversal and Factor-Reversal tests. Cost of living index numbers-family budget and aggregate expenditure methods. An introduction to Whole sale Price Index and Consumer Price Index.

PROBABILITY

Random experiments. Complement, union and intersection of events and their meaning, Mutually exclusive, equally likely and Independent events. Classical, frequency and Axiomatic approaches to probability. Monotone property, Addition theorem (up to 3 events) .Conditional probability. Multiplication theorem (up to 3 events). Independence of events. Baye’s theorem.

PROBABILITY DISTRIBUTION OF UNIVARIATE RANDOM VARIABLES

Concept of random variables. Discrete and continuous random variables. Probability mass and density functions and cumulative distribution functions. Evaluation of conditional probabilities. Evaluation of unconditional probabilities. Change of variables- methods of Jacobian and cumulative distribution function (one variable case).

CO2

PROBABILITY DISTRIBUTION OF BIVARIATE RANDOM VARIABLES

Concept of a two-component random vector. Bivariate probability mass and density functions. Marginal and conditional distributions. Independence of Bivariate random variables.

CORRELATION AND REGRESSION

Bivariate data types of correlation. Scatter diagram. Karl Pearson’s product- moment and spearman’s rank correlations coefficients. Regression equations- fitting of polynomial equations of degree one and two; exponential curve. Two types of regression curves, Identification of regression equations.

Expectation of random variables and their functions. Definition of – Raw moments, central moments and their interrelation, A.M, G.M, H.M, S.D, M.D., covariance, Pearson’s correlation coefficient in terms of expectation. MGF and characteristic function and simple properties. Moments from mgf.

Uniform (discrete/continuous), Bernoulli, binomial, Poisson, Geometric, hyper-geometric, exponential, gamma- one and two parameter(s), beta (type I and type II) – mean, variance, mgf, additive property, lack of memory property. Normal distribution with all properties.

Chebychev’s inequality. Weak Law of Large Numbers -Bernoulli’s and chebychev’s form. Central Limit Theorem (Lindberg- Levy form with proof)

Concept of sampling from a probability distribution. i.i.d observations. Concept of sampling distributions. Statistic(s) and standard error(s). Mean and variance of sample mean when sampling is from a finite population. Sampling distribution of mean and variance from Normal distribution. Chi-square distributions. Student’s   t distribution. Snedecor’s F distribution and statistics following these distributions. Relation among Normal, Chi-square, t and F distributions.

Concepts of Estimation, Estimators and Estimates. Point estimation and Interval estimation. Properties of good estimators; unbiasedness, Efficiency, Consistency and Sufficiency. Factorization theorem (statement).

Method of moments. Method of maximum likelihood. Invariance property of ML Estimators. Method of minimum variance. Cramer-Rao inequality (statement only) confidence intervals for mean, variance and proportions.

Statistical hypotheses, null and alternate hypotheses. Simple and composite hypotheses, Type-I and type-II errors. Critical Region. Size and power of a test, p-value. Neyman-Pearson approach. Large sample tests – z-tests for means, z-tests for difference of means, z-tests for proportion, z-tests for difference of proportion. Chi-square tests for independence, homogeneity.

Normal tests for mean, (when o known). Normal tests for difference of means (when o known). Normal tests for proportion (when o known). t-tests for means (when o unknown), t-tests for difference of means (when o unknown), paired t-test ,test for proportion(binomial), chi-square test. F-test for ratio of variances.

Introduction to Statistics-Introduction to Statistics. Need and importance of Statistics in Psychology. Variables and attributes, Levels of Measurement: Nominal, Ordinal, Interval and Ratio. Collection of data-primary and secondary, census and sampling, classification and tabulation, grouped and ungrouped frequency table. Diagrammatical and graphical representation of data- bar diagram, pie diagram, frequency polygon and curve, histogram, ogives.

Census and Sampling. Different methods of sampling. Requisites of a good sampling method. Advantages of sampling methods. Simple random sampling, Stratified sampling. Systematic sampling.

Measures of central tendency-mean, median and mode- properties, merits and Demerits.

Measures of dispersion-Range, quartile deviation, Mean deviation, standard deviation-properties, merits and demerits, coefficient of variation.

Raw Moments, Central Moments, Inter Relationships (First Four Moments), Skewness – Measures – Pearson, Bowley and Moment Measure, Kurtosis- Measures of Kurtosis – Moment Measure.

Karl Pearson’s Coefficient of Correlation, Scatter Diagram, Interpretation of Correlation Coefficient, Rank Correlation, Regression, Regression Equation, Identifying the Regression Lines.

Probability: Basic concepts, different approaches, conditional probability, independence, addition theorem, multiplication theorem (without proof) for two events, simple examples.

Random variables, Discrete and Continuous, p.m.f and p.d.f., c.d.f of discrete r.v. Mathematical Expectation of a discrete r.v, Mean and Variance of a discrete r.v.

Binomial distribution- mean and variance, simple examples. Normal distribution -definition, p.d.f, simple properties, calculation of probabilities using standard normal tables, simple problems.

Testing of hypothesis- Statistical hypothesis, Simple and composite hypothesis Null and Alternate hypothesis, Type I and Type II errors, Critical Region, Size of the test, P value.

Large sample tests – z-tests for means, difference of means, proportion and difference of proportion, chi-square tests for independence, homogeneity.

Normal tests for mean, difference of means and proportion (when σ known), t-tests for mean and difference of means (when σ unknown), paired t-test, test for proportion (binomial), chi – square test for variance, F-test for ratio of variances.

Definitions of vector and matrix. Types of matrices, Addition, subtraction and multiplication of matrices. Determinants, Minors, Cofactors, Adjoint and Inverse of a matrix. Solution of a system of linear equations – Cramer’s rule and Inversion method. Rank of a matrix -Linear independence of vectors. Some applications in Economics – Input -output analysis – Partial equilibrium market model.

Limit of a function – Derivative of a function. Rules of differentiation – Higher order derivatives – L’Hospital rule of finding the limit of a function. Differentiation of implicit function – Partial and total derivative of a function with several variables. Maxima and minima of a function. Curvature properties – Convexity and concavity – Points of inflection. Properties of homogeneous functions – Euler’s theorem. Matrix calculus: Rules of Matrix differentiation, differentiation of a matrix by a scalar, differentiation of a scalar by a matrix. Some applications in Economics- Derivation of Marginal cost, Marginal revenue functions – Derivation of point elasticity, tax yield and income multiplier, problems relating to indifference curve and isoquant. Production function, utility functions, cost functions. CobbDouglas production function, CES production function – Comparative static analysis of market model, national income model, input output model, determination of partial elasticities of demand.

Indefinite integrals – rules of integration, initial conditions and boundary conditions. Integration by substitution, Integration by parts – Integration of natural exponential functions. Definite integrals – properties of definite integrals. Area under a curve, area between curves. Difference equations and differential equations (basic concepts only). Improper integrals – Beta and Gamma integrals. Some applications in Economics – Consumer surplus and producer surplus – continuous interest-discount calculation. Cobweb model, multiplier accelerator. Harrod – Domar and Solow model.

Formulation of LPP and solution using graphical and Simplex methods. Duality theory – constrained optimization with inequality and non-negativity constrains. Kuhn-Tucker formulation. Primal and duel, shadow prices. Applications from Economics and Finance.

Definitions of vector and matrix. Types of matrices, Addition, subtraction and multiplication of matrices. Determinants, Minors, Cofactors, Adjoint and Inverse of a matrix. Solution of a system of linear equations – Cramer’s rule and Inversion method. Rank of a matrix -Linear independence of vectors. Some applications in Economics – Input -output analysis – Partial equilibrium market model.

Limit of a function – Derivative of a function. Rules of differentiation – Higher order derivatives – L’Hospital rule of finding the limit of a function. Differentiation of implicit function – Partial and total derivative of a function with several variables. Maxima and minima of a function. Curvature properties – Convexity and concavity – Points of inflection. Properties of homogeneous functions – Euler’s theorem. Matrix calculus: Rules of Matrix differentiation, differentiation of a matrix by a scalar, differentiation of a scalar by a matrix. Some applications in Economics- Derivation of Marginal cost, Marginal revenue functions – Derivation of point elasticity, tax yield and income multiplier, problems relating to indifference curve and isoquant. Production function, utility functions, cost functions. CobbDouglas production function, CES production function – Comparative static analysis of market model, national income model, input output model, determination of partial elasticities of demand.

Indefinite integrals – rules of integration, initial conditions and boundary conditions. Integration by substitution, Integration by parts – Integration of natural exponential functions. Definite integrals – properties of definite integrals. Area under a curve, area between curves. Difference equations and differential equations (basic concepts only). Improper integrals – Beta and Gamma integrals. Some applications in Economics – Consumer surplus and producer surplus – continuous interest-discount calculation. Cobweb model, multiplier accelerator. Harrod – Domar and Solow model.

Linear Programming

Formulation of LPP and solution using graphical and Simplex methods. Duality theory – constrained optimization with inequality and non-negativity constrains. Kuhn-Tucker formulation. Primal and duel, shadow prices. Applications from Economics and Finance.