Probability and Mathematical Statistics 1

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Code Completion Credits Range Language
01PRA1 Z,ZK 6 4+2 Czech
Department of Mathematics

The subject is devoted to the introduction to Theory of probability and statistics on measure-theoretic level for discrete models, continuous distributions and general distributions of random variables. We deal with sample an integral characteristics of random variables and variants of limit theorems are derived (LLN, CLT). This knowledge is further applied to the statistical processing of observations and statistical parametric model estimation.


Basic course of Calculus (in the extent of the courses 01MAA3-4 or 01MAB3-4 held at the FNSPE CTU in Prague).

Syllabus of lectures:

Axioms of probability space, sigma-fields, probability measure. Dependent and independent events.Borel sets, measurable functions, random variables and probability distributions. Radon-Nikodym theorem. Discrete and absolutely continuous distributions, examples. Product measure, integral w.r.t. probability measure. Expectation of random variables, moments and central moments. Lp space, Schwarz inequality, Chebyshev inequality, covariance. Characteristic function and its properties, applications. Almost sure convergence, in Lp, convergence in probability. Law of large numbers (Chebyshev, Kolmogorov,...). Weak convergence, its properties, Lévy theorem, Slutsky lemma, central limit theorems (CLT), Lindeberg-Feller fundamental CLT, Lindeberg condition, Berry-Esseen theorem. The multivariate normal distribution with its properties. Cochran's theorem and the independence of the sample mean and sample variance. Introduction to statistical inference, populations, natural extensions in sample space, the existence of independently distributed sequences of observations. The problem of point statistical estimation, parametric and nonparametric caase, optimality criteria, asymptotic normality. Sample moments and other empirical characteristics.

Syllabus of tutorials:

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, Exponencial,...). 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 Corelation of selected random variables. 8. Law of large numbers and Central limit theorems - asymptotics and usefullness. 9. Two dimensional normal distribution 10. Point statistical estimation - examples of sample estimates and their consistency and unbiasedness, sample moments.

Study Objective:

Knowledge: In frame of the basic course in Probability and Statistics 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.

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

Study materials:


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

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


[3] Shao J., Mathematical Statistics, Springer, 1999.

[4] Lehmann E.L., Point Estimation, Wiley, N.Y., 1984.

[5] Lehmann E.L., Testing Statistical Hypotheses, Springer, N.Y., 1986.

Further information:
No time-table has been prepared for this course
The course is a part of the following study plans:
Data valid to 2023-02-02
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