Probability theory and its applications - free course from Open Education, training 5 weeks, from 8 to 10 hours per week, Date: December 3, 2023.
Miscellaneous / / December 07, 2023
Position: Academic director of the educational program "Computer Science and Data Analysis"
1. Classical and discrete probability
We will begin our study of probability theory with a natural question: how do we understand what probability is? In the first week, we will understand probability as the frequency with which an event occurs. To develop an understanding of the basic principles of probability and get started quickly, we will need a powerful tool - the concept of an event tree. At first we will use it without strict justification, but understanding the principle of operation.
In the second week we will justify the event tree using a more advanced technique. Without further delay, we will introduce the most commonly used concept in probability theory: the random variable. We immediately use this concept to work with the standard model - the Bernoulli scheme. The week ends with the Poisson distribution, which is closely related to the Bernoulli scheme. The Poisson distribution is used to describe the flow of requests from queuing systems. So by the end of the first week you will have a rich set of examples of using probabilistic models in practice.
2. Conditional probability and independence
The concept of “conditional probability” will be related to the material of the second week. We will study how events are interconnected. To use information about the relationship of events, use the multiplication theorems and the total probability formula, which will be formulated in the middle of the week. Continuous random variable
Up to this point, we have not yet considered probability spaces in which each individual outcome has zero probability. This week we will learn how we can define and use continuous random variables. Axiomatics A will serve as our theoretical foundation. N. Kolmogorov, a great mathematician and founder of modern probability theory.
3. Expected value
Most objects that need to be analyzed are described by a random variable. But how to evaluate the random variable itself? One of the most important numerical characteristics of a random variable is its mathematical expectation. Moreover, it turns out that in some situations, knowledge of the mathematical expectation allows one to estimate the values of a random variable and make extremely useful observations. It is this section of science that the third part of our studies will be devoted to.
4. Variance and Covariance
Let's learn about the meaning of the variance of a random variable, which allows us to conduct a much more accurate analysis of the situation. In addition, we will learn which methods allow us to estimate the dependence between random variables.