Academic year/semester: 2024/25/2

ECTS Credits: 4

Available for: All OU students

Lecture hours: 2
Seminarium:1
Practice:
Laboratory: 0
Consultation: 0

Prerequisites:

Course Leader: Dr. KÁRÁSZ Péter

Faculty: John von Neumann Faculty of Informatics, 1034 Budapest, Bécsi út 96/b

Course Description:
Goal: To lay the foundations of probability theory and statistics
Course description: Kolmogorov probability space; law of total probability; conditional probability; Bayes’ theorem; probability distribution function; expectation, variance and moments; special distributions (Poisson, uniform, etc.). Moment generating function, characteristic function. Joint distributions; random vectors; independence; covariance matrix. General definition and properties of conditional expectation; law of total expectation. Types of convergence; Borel-Cantelli lemmas; laws of large numbers; sums of random variables; central limit theorems. Statistical space; sample; statistics; ordered sample; empirical distribution function; Glivenko-Cantelli theorem. Estimation techniques, maximum-likelihood estimation, method of moments, method of least squares. Hypothesis testing; confidence intervals. Parametric and nonparametric tests.

Competences:

Topics:
1. Kolmogorov probability space and related notions. Examples.
2. Law of total probability; conditional probability, Bayes’ theorem. Random variables and their properties. Probability distribution function; expectation, variance and moments
3. Special discrete and continuous random variables and their properties (Poisson, uniform distributions, etc.)
4. Continuation of lecture 3 plus moment generating functions, characteristic function
5. Joint distributions; random vectors; independence; covariance matrix.
6. General definition and properties of conditional expectation; law of total expectation.
7. Types of convergence; Borel-Cantelli lemmas; laws of large numbers; sums of random variables; central limit theorems.
8. Continuation of lecture 7.
9. Statistical space; sample; statistics; ordered sample; empirical distribution function; Glivenko-Cantelli theorem.
10. Continuation of lecture 9.
11. Estimation techniques, maximum-likelihood estimation, method of moments, method of least squares.
12. Hypothesis testing; confidence intervals
13. Parametric and nonparametric tests
14. Summary

Assessment: Written exam

Exam Types:

Written Exam

Compulsory bibliography: https://www.math.ucdavis.edu/~gravner/MAT135A/resources/lecturenotes.pdf

Recommended bibliography: Gut, A.: An Intermediate Course of Probability, 2nd ed.; Springer; 2009. Gut, A.: Probability: A Graduate Course; Springer; 2005.

Additional bibliography: Lecture notes uploaded to the e-learning system of the university

Additional Information: