Measure and Probability

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Code Completion Credits Range Language
01MIP Z,ZK 6 4+2 Czech
Garant předmětu:
Department of Mathematics

The subject is devoted to the introduction to Theory of probability on measure-theoretic level for discrete models, continuous distributions and general distributions of random variables. We deal with the examples of distributions including multi-dimensional Gaussian distribution and their properties. Further the (non)+integral characteristics of random variables (E, Var,...), convergence modes (Lp, P, a.s., D) and variants of limit theorems are derived (LLN, CLT).


01MAA3-4 or 01MAAB3-4.

Syllabus of lectures:

1. Axioms of probability space, sigma-fields, probability measure.

2. Dependent and independent events. Borel sets, measurable functions, random variables and probability distributions.

3. Radon-Nikodym theorem. Discrete and absolutely continuous distributions, examples.

4. Product measure, integral w.r.t. probability measure.

5. Expectation of random variables, moments and central moments.

6. Lp space, Schwarz inequality, Chebyshev inequality, covariance.

7. Characteristic function and its properties, applications.

8. Almost sure convergence, in Lp, convergence in probability.

9. Law of large numbers (Chebyshev, Kolmogorov,...).

10. Weak convergence, its properties, Lévy theorem, Slutsky lemma.

11. Central limit theorems (CLT), Lindeberg-Feller fundamental CLT, Lindeberg condition, Berry-Esseen theorem.

12. The multivariate normal distribution with its properties.

13. Cochran's theorem and the independence of the sample mean and sample variance, populations, natural extensions in sample space, the existence of independently distributed sequences.

Syllabus of tutorials:

Tasks solvation in the areas:

1. Axioms of probability space

2. Dependent and independent events.

3. Particular discrete distributions, examples (Binomial, Poisson, Pascal, Geometric, Hypergeometric, Multinomial distribution).

4. Particular absolute continuous distributions, examples (Uniform, Gamma, Beta, Normal, Exponential,...).

5. Distributions based on transformations (Student, Chi-squared, Fisher-Snedecer) and quantiles.

6. Computations of characteristic functions, expectations and moments of particular distributions.

7. Covariance and Correlation of selected random variables.

8. Law of large numbers and Central limit theorems - asymptotics and usefulness.

9. Two dimensional normal distribution.

Study Objective:


In frame of the basic course in Probability on measure-theoretic level, to provide students with the knowledge necessary for the following future subjects using probability and stochastic models. To give a deeper insight into the field.


Orientation in majority of standard notions of the probability theory and capabilities of practical applications in actual probabilistic computation.

Study materials:

Key references:

[1] Rényi A., Foundations of probability, Holden-Day Inc., San Francisco, 1970.

[2] Taylor J.C., An Introduction to Measure and Probability, Springer, 1997.

Recommended references:

[3] Jacod J., Protter P., Probability Essentials, Springer, 2000.

[4] Schervish M.J., Theory of Statistics, Springer, 1995.

Further information:
No time-table has been prepared for this course
The course is a part of the following study plans:
Data valid to 2024-06-16
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