Measure and Probability
Code  Completion  Credits  Range  Language 

01MIP  Z,ZK  6  4+2  Czech 
 Garant předmětu:
 Lecturer:
 Tutor:
 Supervisor:
 Department of Mathematics
 Synopsis:

The subject is devoted to the introduction to Theory of probability on measuretheoretic level for discrete models, continuous distributions and general distributions of random variables. We deal with the examples of distributions including multidimensional 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).
 Requirements:

01MAA34 or 01MAAB34.
 Syllabus of lectures:

1. Axioms of probability space, sigmafields, probability measure.
2. Dependent and independent events. Borel sets, measurable functions, random variables and probability distributions.
3. RadonNikodym 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), LindebergFeller fundamental CLT, Lindeberg condition, BerryEsseen 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, Chisquared, FisherSnedecer) 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:

Knowledge:
In frame of the basic course in Probability on measuretheoretic 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.
Skills:
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, HoldenDay 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.
 Note:
 Further information:
 No timetable has been prepared for this course
 The course is a part of the following study plans:

 BS Matematické inženýrství  Matematické modelování (compulsory course of the specialization)
 BS Matematické inženýrství  Aplikované matematickostochastické metody (compulsory course of the specialization)
 Aplikovaná algebra a analýza (compulsory course in the program)
 Aplikované matematickostochastické metody (compulsory course in the program)
 Matematické inženýrství  Matematické modelování (PS)