Probability theory

obligatory courses

The course is an introduction to the basic concepts and methods of the probability theory. The material includes the notion of probability, Kolmogorov’s axioms, conditional probability and independence, overview of basic discrete models of probability theory, basic discrete and continuous probability distributions, parameters of probability distributions, laws of large numbers, Central Limit Theorem, Markov chains, elements of information theory.

1. Probabilistic models of experiments, Kolmogorov’s axioms.

2. Basic combinatorial schemes, classical probability, geometric probability.

3. Conditional probability.

4. Independence of events, Bernoulli trials.

5. Random variables and random vectors, discrete and continuous; distribution (law) of a random vector, cumulative distribution function, probability density function.

6. Parameters of probability distributions: expected value, variance, moments, median, quantiles, covariance matrix.

7. Independence of random variables, criteria of independence for discrete and continuous random variables. Distribution of a sum of independent random variables, convolution of measures.

8. Overview of basic probability distributions.

9. Basic probabilistic inequalities.

10. Laws of large numbers: weak law of large numbers, Kolmogorov’s Strong Law of Large Numbers.

11. Central Limit Theorem.

12. Markov chains. Classification of states. Ergodic theorem for Markov chains.

13. Elements of information theory: Shannon’s entropy, mutual information, interpretation and connections with coding theory.

Rachunek Prawdopodobieństwa dla (Prawie) Każdego - Jakubowski Jacek, Sztencel Rafał, SCRIPT Wydawnictwo, 2006 [in Polish]

After completing the course, the student:

• knows the basic concepts and methods of probability theory: Kolmogorov's axiomatics, conditional probability, independence, continuous and discrete distributions, distribution parameters, laws of large numbers and the central limit theorem, Markov chains, elements of information theory

• is able to understand the basic probabilistic arguments used in the literature related to bioinformatics

• understands the nature of probabilistic modeling of natural phenomena

• can build and analyze probabilistic models of simple random phenomena, using the basic tools and theorems of the probability theory

• is ready to further study the theory of statistics and data processing

written exam