Probability Theory and Statistics

1.      Historical introduction. Elementary combinatorics: permutation, combination, variation.

Basic concepts: random experiment, event space, event, elementary event, operation between events, axioms

2.      Properties of probability: Poincare-rule, Boole’s inequalities, classical and geometrical probability field

3.      Conditonial probability, independency of events, Theorem of total probability, Bayes’s Theorem, produce theory

4.   Random variable, probability distribution function, discrete distribution, expected value. Binomial, Poisson, geometrical distribution. Approximation of the Poisson distribution by the binomial distribution.

5.   Continouos cases, properties of the distribution function, probability density function, expected value, transformation of random variables.

6.   Notable continouos distributions: uniform, exponential. Simulation with uniform distribution. Memoriless property of the exponential and the geometric distribution. Variance, moments.

7.    Joint distribution function, projective distribution functions. Independency. Joint density function, projective density function. Covariance, correlation.

8.    Conditional distribution, conditional expectation (regression). Linear regression. Properties of regression. Examples of discrete and continouos cases.

9.   Basics of mathematical statistics: sample, parameter, statistics. Properties of estimation: unbiased, consistency, efficiency

10. Estimation of average and variance, maximum likelihood, method of moments, nonparametric methods, empirical distribution function, regression estimation

11.  Theorems of large numbers, Markov’s- and Chebisev’s inequalities.

12.  Normal distribution, standardization. Central limit theorems, Moivre-Laplace’s Theorem

13.  Two-dimensional normal distribution. Connection of the independency and uncorrelatedness in normal case. Regression in case of normal distribution.