Courses on probability theory - course RUB 24,475. from Online school TutorOnline, training 55 ac. hours, Date: December 2, 2023.

Probability theory

Topic 1. Random events - 23 hours.

1. Subject of probability theory.
2. The importance of statistical methods.
3. Statistical approach to describing random phenomena.
4. The concept of a random event.
5. Space of elementary events, frequency of events, reliable, impossible and random events.
6. Composite events, actions on events.
7. Algebra of events as one of the interpretations of Boole algebra.
8. Venn diagrams
9. Classical and statistical definition of probability, geometric probability.
10. The limitations of the classical and statistical definitions of probability, geometric probability in describing real phenomena.
11. Event field.
12. Axiomatic definition of probability.
13. Basic combinatorial objects: permutations, placements, combinations, partitions.
14. Using combinatorics methods in probability theory.
15. Properties of probability.
16. Conditional probability.
17. Independent events.
18. Probability addition and multiplication theorems.
19. Total probability formula and Bayes formula.
20. Repetition of Bernoulli's tests.
21. Local and integral theorems of Laplace.
22. Deviation of relative frequency from constant probability in independent trials.
23. The most likely number of occurrences of an event in independent trials.

Topic 2. Random variables - 25 hours.

1. Discrete random variables.
2. Distribution law of a discrete random variable.
3. Distribution polygon.
4. Cumulative distribution function and its properties.
5. Probability distribution density.
6. Numerical characteristics of random variables (mathematical expectation, variance, mean square deviation, initial and central moments, mode, median, skewness and kurtosis coefficients) and their properties.
7. Mathematical expectation and dispersion, their properties.
8. Moments of random variables.
9. Examples of distribution laws for discrete and continuous random variables.
10. Distribution of functions of random arguments.
11. Binomial distribution, Poisson distribution.
12. System of two random variables.
13. The law of probability distribution of a discrete two-dimensional quantity.
14. Function and density of distribution, their properties.
15. Continuous random variables.
16. Distribution density function and its properties.
17. Relationship between differential and integral distribution functions.
18. Uniform, normal, exponential distribution.
19. Conditional laws of distribution of components of two-dimensional quantities.
20. Conditional mathematical expectation.
21. Necessary and sufficient conditions for the independence of random variables.
22. Numerical characteristics of a system of two random variables.
23. Correlation moment and correlation coefficient.
24. Generalization of two-dimensional random variables to n-dimensional variables.
25. Regression functions.

Topic 3. Limit theorems of probability theory - 7 hours.

1. Mass phenomena and the law of large numbers.
2. Chebyshev's inequality.
3. Chebyshev's theorem and its significance for practice.
4. Central limit theorem.
5. Bernoulli's theorem
6. De Moivre-Laplace theorem.
7. Poisson's theorem.