Probability Calculus

obligatory courses

The objective of the course is to introduce the basic topics of probability calculus, in the extent it may be of use for economists (in statistics, econometrics).

The content includes the foundations of probability theory, random variables and limit theorems, i.e. topics essential for understanding statistic and econometric methods and models.

Assessment: written exam (problems to solve)

Prerequisite: Calculus

The course consists of lectures and discussions. The discussions will be devoted to solving practical problems related to theoretical issues introduced during lectures. The topics covered include

1. Elementary probability calculus

a) sample spaces (discrete, continuous)

b) Kolmogorov's axioms

c) basic properties of probability

d) conditional probability

e) Bayes' theorem

f) independent events

g) Bernoulli Scheme

h) Poisson theorem

2. Random variables and distributions

a) definition and concept of random variables (discrete and continuous)

b) probability distribution functions (discrete and continuous)

c) definition, concept and properties of probability density

d) properties of cumulative distribution functions

e) quantiles

f) characteristics of random variables (expected value, variance, covariance, moments), most common distributions

g) sample characteristics (mean, variance, etc.)

h) joint probability distribution functions, joint probability densities

i) Schwarz inequality, Chebyshev inequality, Bernstein inequality

j) expected value of a random variable function (characteristic and moment generating functions)

k) independence of random variables, its consequences and criteria

l) distributions of sums of random variables. Gamma, Chi-square, F distributions

m) conditional expectation (for discrete and continuous random variables)

n) two- and multi-dimensional normal distributions

3. Limit theorems

a) strong and weak Laws of Large Numbers (specific cases) and applications

b) de Moivre-Laplace theorem and applications

c) Central Limit Thoeroem and applications

4. Introduction to Markov Chains

Lecture notes (to be distributed)

Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, available online

Sheldon M. Ross, Introduction to Probability Models, available in the FoES library and online

Wackerly, D., Mendenhall, W., & Scheaffer, R. Mathematical statistics with applications, available in the FoES library

Upon the completion of the course, the student:

-- knows and understands the basic concepts and theorems of probability calculus, which are used in statistics, econometrics, insurance theory and mathematical choice models -- random experiment (continuous and discrete models), probability, conditional probability, Bayes’ theorem, independence of events (X1A_W01, X1A_W03, S1A_W06)

-- has the ability to describe random occurrences with the use of a formal language – mathematics, and knows the difference between continuous and discrete random variables and how to describe each of them (in terms of a cumulative distribution function, probability density etc.) (X1A_U01)

-- knows how to apply the basic techniques of probability calculus to solve problems, e.g. economic – from insurance theory, financial markets, microeconomics; in particular the student

-- is able to construct a model of a random experiment

-- is able to solve simple “descriptive” problems

-- is able to determine the distributions of random variables corresponding to simple experiments and calculate their characteristics (such as means, variances, quantiles etc.)

-- is able to find marginal distributions of multi-dimensional random variables, their characteristics and correlation, as well as a linear approximation, based on a joint distribution

-- is able to calculate the conditional density function and the conditional expected value and solve simple descriptive problems where these concepts appear

-- knows how to use the conditional random variable for approximation and forecasting

-- is able to solve basic problems with the use of limit theorems

-- is able to construct a Markov chain and use it to find the solution to a simple problem

(S1A_U07)

-- knows how to interpret the results obtained when modeling the economy and infer conclusions; in particular, the student is aware of the existence of various types of convergence o random variables and limit theorems and their consequences (S1A_U07)

-- is aware of the applications of probability calculus in economics in general, and statistics and econometrics in particular

-- is capable of acting logically and accurately

(S1A_K01)

Tutorials: group activity during classes (presentation of problem solutions 20%) +individual activity on the moodle platform (20%) + two tests (problem solving tasks, 30% each)

Final grade for the course: result from classes (40%) + final exam (60%)

Final exam: written exam (5-7 problems to solve), in the case of an online session an oral exam is possible.

Attendance during classes is mandatory.