Probability theory I

obligatory courses

(in Polish) Oczekuje się dobrej znajomości zagadnień ujętych w sylabusie przedmiotu Analiza matematyczna II.1.

Kolmogorov axioms. Basic probabilities.

Random variables, probability distributions, and their parameters. Independence.

Convergence of random variables. Basic limit theorems: Poisson theorem, weak and strong laws of large numbers, de Moivre-Laplace theorem.

Kolmogorov axioms. Properties of probability measures. Borel-Cantelli lemma. Conditional probability. Bayes' theorem.

Basic probabilities: classical probability, discrete probability, geometric probability.

Random variables (one- and multidimensional), their distributions. Distribution functions.

Discrete and continuous distributions. Distribution densities. Parameters of distributions: mean value, variance, covariance. Chebyshev inequality.

Independence of: events, sigma-fields, random variables. Bernoulli (binomial) process.

Poisson theorem. Distrubution of sums of independent random variables.

Convergence of random variables. Laws of large numbers: weak and strong. De Moivre-Laplace theorem.

Billingsley, P., Probability and Measure.

Feller, W., An introduction to probability theory and its applications. vol. I, II,

Shiryayev, A. N., Probability, New York : Springer-Verlag, 1984.

1. Student are familiar with the notion of probability space and understand its role in the mathematical description of random phenomena.

2. Students are able to solve combinatorial problems related to counting

3. Students know the notion of conditional probability and can apply the total probability and Bayes' laws.

4. Students know the definitions of independence of events and sigma-algebras and understand the difference between joint and pairwise independence.

5. Students know the definition of a random variable and its law. They can infer basic properties of the law from the probability distribution function.

6. Students know various techniques of identifying the law of random variables and verifying their independence.

7. Students are familiar with basic examples of discrete and continuous probability distributions. They are able to list examples of random phenomena that can be modelled using such distributions.

8. Students know the notions of expectation, variance and covariance. They can compute parameters of random variables and know the relation between independence and lack of correlation.

9. Students are able to verify convergence of sequences of random variables. They know relations between various modes of convergence (almost sure convergence, convergence in probability and in L^p) and can illustrate them with examples.

10. Student can formulate the strong law of large numbers and provide examples of applications.

11. Students are familiar with the de Moivre-Laplace theorem and can apply it to approximate probabilities of appropriate events.

he final grade is based on the number of points gained during classes, the midterm exam and the final exam.