Math203
Probability Theory 1
Faculty
Andrey Khokhlov
Chief Researcher, IEPT RAS
Course length
Duration
Total hours
Credits
Language
Course type
Fee for single course
Fee for degree students
Skills you’ll learn
Overview
The course is aimed to develop an in depth familiarity with discrete probability models, corresponding methods of calculations and motivation of consideration of limiting cases (i.e. non-discrete distributions). Beginners need to start with a stock of practical examples (explaining why the corresponding mathematical theory is needed) before they can proceed to the analysis of important examples with a detailed explanation of new concepts and methods of calculations (including those relating to computers). This will take approximately two-thirds of the course’s time.
We will start with classical examples, in which the mathematical model of outcomes has symmetries, and we will pay increased attention to non-trivial combinatorial methods for the case of distinguishable or indistinguishable objects. Standard concepts of elementary probability theory further generate several versions: Bayesian methods, frequency approach, measure theory, quantum probabilities --- will be parsed simple examples in each of these cases.
The last part of the course is devoted to random variables associated with the previously considered examples. We will consider important discrete distributions in probability theory with some results of their limiting behaviour. From the classical results, we will analyse the Law of Large Numbers and discuss its practical meaning for Mathematical Statistics.
Learning highlights
- Learning objectives: to introduce the list of probabilistic examples practically important for applications and to build further theory. Using these examples and the so-called "paradoxes of probability theory".
- Indicate reliable methods of mathematical model construction for practical situations, prepare the basis for understanding mathematical statistics methods.
- Teach basic terminology in the field of probabilistic methods and point out some differences in terminology typical for machine learning and pattern recognition tasks.
- Introduce standard programs and teach computer modelling of some processes using randomness.
Course outline
15 classes
Class 1
Probability considerations and formal descriptions. The empirical background and agreements. What is a probability model anyway?
Class 2
The sample space. Finite sets. Events and relations among events. Probabilities in discrete sample spaces: preparations.
Class 3
Elements of combinatorial analysis: ordering, partitioning. Basic formulas.
Class 4
Finite sets considerations: basic combinatorial formulas and applications. Binomial coefficients. Random walk model.
Class 5
Combinatorics of distinguishable and indistinguishable objects. Methods of computation: the generating functions and recurrence relations
Class 6
Discrete models in probability, the use of symmetry. Axiomatic discrete probabilities and paradoxes.
Class 7
Non-discrete geometrical probabilistic models and the use of symmetry. Multi-dimensional features and computer examples.
Class 8
Number sets and vector spaces as Probability Spaces. Sequence of measurements in Natural Sciences and the probability model. What is Mathematical Statistics anyway?
Class 9
Computer examples, random number generator(s). What is pseudo-random generator anyway? Computer experiments.
Class 10
Conditional probabilities and the Bayesian approach. Midterm test.
Class 11
Random variables --- definition and examples. Examples and terminology.
Class 12
Discrete random variables and their distributions. Expectation and other numerical characteristics.
Class 13
Some important discrete distributions and their generating functions. Computations with random variables and random vectors.
Class 14
Sequences of random variables. The concepts of convergence in probability theory. Some examples of the non-discrete random variables.
Class 15
The Chebyshev inequality and the Law of Large Numbers. Concluding discussion and final test.
Prerequisites
The elementary sections of analytical geometry, mathematical analysis, and linear algebra are obligatory.
In particular, skill to calculate areas and volumes of the simplest geometrical figures, acquaintance to the concept of multidimensional euclidean space, matrix algebra. It is also necessary to be able to calculate the simple limits of sequences and expand elementary functions in power series.
Computer methods are required: to write a simple (3-4 lines) program in Python or similar language (Matlab, Octave), in particular, to plot a function given either by an explicit formula or by a set of argument-value pairs (also for functions of several variables).
Methodology
While studying examples in detail to analyse initial requirements to probability model, combinatorial methods of calculations, computer implementation of used methods and corresponding terminology. Build a hierarchy of examples with the purpose of their further use in advanced chapters of Probability Theory. Homework should reinforce the material and identify possible difficulties in understanding.
Grading
After getting his Ph.D. in Algebraic Topology in 1983 Andrey worked in several scientific and/or teaching organisations, among them are the Russian Academy of Sciences, Moscow State University, and Baumann Technology University. The Scientific advising of the graduate and thesis students was part of his activities, not only in Russia, but also in France.
Andrey’s main results in science are linked with geophysical data processing, so naturally his teaching interests are now concentrated in the applied methods of Statistics and their algorithmic implementations. He currently helps his students avoid some common errors within the probabilistic inferences and support their attempts to study Probability and Statistics theory in general.
See full profileApply for this course
Probability Theory 1
by Andrey Khokhlov
Total hours
45 Hours
Dates
Nov 09 - Nov 27, 2020
Fee for single course
€1500
Fee for degree students
€750
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FAQ
Will I receive a certificate after completion?
Yes. Upon completion of the course, you will receive a certificate signed by the director of the program your course belonged to.
Do I need a visa?
This depends on your case. Please check with the Spanish or Thai consulate in your country of residence about visa requirements. We will do our part to provide you with the necessary documents, such as the Certificate of Enrollment.
Can I get a discount?
Yes. The easiest way to enroll in a course at a discounted price is to register for multiple courses. Registering for multiple courses will reduce the cost per individual course. Please ask the Admissions Office for more information about the other kinds of discounts we offer and what you can do to receive one.