PROBABILITY THEORY AND MATHEMATICAL STATISTICS
(56 hours of lectures and 28 hours of seminars)
Author: Dr. Prof. Serguei Aivazian

1-2 modules, 2002 / 2003

Lectures:
Probability Theory: Serguei Aivazian, Pavel Katyshev
Mathematical Statistics: Serguei Aivazian, Anatoly Peresetsky

TAs: D.Sokolov, T.Belkina, S.Golovan, V.Kazakov

The purpose and brief description of the course. The methods and models of special sample analysis, study of time series and systems of econometric equations, production functions, demand and supply functions, probabilistic models of economic growth and equilibrium, multidimensional statistical analysis of economic information, mathematical models of insurance, markov models of population movement play important role among different techniques used in social and economic studies. It is impossible to understand and to use properly these techniques without the adequate sound knowledge of probability theory and mathematical statistics. This course is aimed to cover the necessary minimum of these techniques. The construction of syllabus takes into account that it is designed for a user of the presented methods and models and hence is aimed to describe its applied possibilities and recommendations for use.

The course consists of two parts. Part I "Probability theory" (12 lectures) covers the basics of mathematical discipline designed to study the properties of the models that imitate the mechanisms of functioning of real (i.e. social and economic) systems which conditions of life include the inevitability of influence of big number of random variables. The formulated above aim stresses the following notions: multidimensional joint and conditional distributions, the markov and regression models. Part II "Mathematical statistics"(16 lectures) covers the main notions, tools, mathematical methods and models destined to organise the collection, systematisation and analysis of statistic data with the aim of its presentation, interpretation and drawing of scientific and practical conclusion. The special attention in this part is paid to the methods of statistic estimation of the unknown parameters of the model and verification of statistical hypotheses.

The course objective. The course is aimed at equipping the students with the skills of probabilistic modeling of real socio-economic processes, economic interpretation of genesis of the analysed data, its applied statistic analysis, construction, identification and verification of statistical models of the analysed phenomena, computer realization of techniques and methods.
The content of the course "Probability theory and mathematical statistics" is an important ingredient of theoretical and methodological base of a number of further courses in econometrics, elected course in risk theory, etc.

The form of control. During lectures and seminars students will have to submit two independent works during each part and there are two written examinations: exam in "Probability theory"(7th week) and exam in "Mathematical statistics"(16th week). The examples of problems for independent and examination works are given in the appendix. The final grade in each part takes into account the results of independent works (with weight 1/3) and examination (with weight 2/3).

Text book: S.Aivazian, V.Mkhitarian; "Applied Statistics and Base of Econometrics".

Brief contents of the course:

PART I: PROBABILITY THEORY

Lecture 1. Introduction in the probability theory and mathematical statistics.
The basic definitions of the probability theory and mathematical statistics. The basic types of socio-economic problems which can be resolved with the help of the methods and models of probability theory and mathematical statistics. The probability theory and the conditions of the statistic ensemble. The main types of real situations from the point of view of the statistic ensemble. The concept of the subjective probabilities. The interrelationships of the probability theory, mathematical statistics and other statistical disciplines.
Lecture 2 and 3. Topic 1. The main concepts of the probability theory
1.1. Discrete probability space. The notion of the random experiment. The random events and the operations over them. The axiomatic introduction of the probabilities of the elementary events and the rules of computation of probability of any event. The notion of the discrete probability space. The theorems of addition and multiplication of the probabilities. The conditional probability. The independence of events. The formula of the complete probability and the formula of Bayes.
1.2. Continuous probability space (A.N.Kolmogorov axiomatics). The specifics of the general (continuous) case of the probability space. The notion of the theoretical-multiple concepts and measure theory and its use in the construction of the measure theory. Random events, their probabilities and operations over them (Kolmogorov's axiomatic approach).
Lecture 4 and 5. Topic 1.The main concepts of the probability theory (continuation)
1.3. Random variable and its main characteristics: Definition, examples and the main types of the random variables. The possible values of the random variables.
Discrete random variable Its probability distribution and main numerical characteristics. Bernoulli trial scheme and binomial law of probability distribution. The notions of partial (marginal) and conditional distributions (on the example of two-dimensional discrete random variable). Independent random variables. Covariance, correlation coefficient and their properties.
Continuous random variable Its probability distribution, density function and main numerical characteristics. Normal (gaussian) law of probability distribution. The notion of multidimensional law in a continuous case. The notions of partial (marginal) and conditional distributions (on the example of two-dimensional normal random variable). The link between the independence of random variables and the value of the correlation coefficient in this case. Conditional mathematical expectation and the regression function.
Lecture 6 and 7. Topic 1.The main concepts of the probability theory (end)
1.4. The laws of probability distribution most widely used in socio-economic applications and their main properties. The mechanism of their formation. Examples. Analytical problems. The graphs and moments of the following distributions: binomial, hypergeometric, Poisson, normal, polinomial, exponential, Weibull, Laplace, Pareto, Cauchy, lognormal. The comments on computer simulation of values of random variable given its distribution.
Lecture 8 and 9. Topic 2. The main results of the probability theory 2.1 The probability distributions for transformations of given random variables. General problem setup and its applied meaning. The probability distribution of a monotonous function of a given random variable. Generalization for a multidimensional case (without proof). The probability distribution of a sum of two independent random variables (the composition formula).
2.2 Tchebyshev Inequality. The problem of estimate of probability of given deviations of its
mean values given its variances. The derivation and interpretation of meaning and exactness of inequality for symmetrically distributed random variables.
2.3 Law of big numbers and its corollary. The law of big numbers as the statement of
property of statistical stability of sampling means. Bernoulli's theorem. Statistical stability
of sampling characteristics. The computer demonstration of the law of big numbers.
2.4 The special role of normal distribution: the central limit theorem. The notion of asymptotic normality of sequence of random variables. The formulation of the central limit theorem for independent identically distributed summands with finite variance (without proof); de Moivre-Laplace theorem about asymptotic normality of binomial random variable (as a consequence of the central limit theorem); an illustration of effects of the central limit theorem on the computer).
Lectures 10, 11, and 12. Topic 1: Markov chains and their use in modelling of social and economic processes. 3.1 Basic concepts and definitions of the Marcov chains theory and a review of their social and economic applications: a definition of a Markov chain, examples; notion of matrix of transition probabilities, stochastic matrix; application of Markov chains for modelling of population movement processes. 3.2. Certain results of the Markov chains theory: classification of states; calculation of probability of transition from one state to another in a given number of steps; stationary distributions and the ergodic property of irreducible chains.

SECTION 2: MATHEMATICAL STATISTICS

Lecture 13. Topic 4: The basis of statistical description. 4.1 General view on the economic research using mathematical and statistical tools. 4.2. Population, sampling, and their basic characteristics: average, dispersion, asymmetry, excess, quantiles (percentage points), distribution and dencity functions; two variants of interpretation of a sampling -- practical and hypotetical. 4.3. Various schemes of sample analysis: simple random sample, stratified sample, and their combinations.
Lecture 14 and 15. Topic 4: The basis of statistical description (continuation). 4.4. Analysis of basic sampling charachteristics behavior: asymptotic behavior of basic sampling charachteristics, their convergency to the respective theoretical values, character of their random variation; sample average and variance behavior under finite sampling; basic distributions connected with normal distribution: "chi-square", t, Ficher (their definition via combination of independent standard normal distributed random variables, tables' using)
Lecture 16. Topic 4: Basics of statistical description (conclusion).
4.5 Variation series and ordinal statistics: definition, one and two ordinal statistics distribution derivation.
Lectures 17-21. Topic 5: Statistical parameter estimation.
5.1. Statistical estimates and their properties: unbiasedness, consistency, efficiency. 5.2. Rao-Kramer-Freshet information inequality and estimate efficiency measurement; examples of "irregular" situations (uniform and exponential with drift distributions). 5.3. The main methods of statistical estimation: Maximum Likelihood, moments, Least Squares; their comparative analysis. 5.4. Construction of interval estimates (general approximate approach and the examples of precise construction). 5.5. Bayesian approach to the statistical estimation.
Lectures 22-25. Topic 6: Basics of the statistical hypothesis testing theory.
6.1. Types of statistical criteria and their application: goodness of fit test (concerning the distribution function), homogeneity, series of observations stationarity, parametric criteria. 6.2. General scheme of any statistical criteria and its quality characteristics. 6.3. Elements of theory and statistical criteria examples : Neumann-Pearson lemma concerning the strongest criterion, concept of sequential procedures.
Lectures 26-28. Topic 7: Elements of regression and variance analysis.
7.1. General scheme of statistical dependence investigation. 7.2. Classical model of simple regression and classical method of least squares (OLS). 7.3. Statistical analysis of simple regression in framework of two-dimensional normal distribution. 7.4. Concept of variation analysis (one- and two-factor models)