Probability theory
Description
The course Probability Theory is designed to familiarize students with the basic concepts of probability theory: random events, classical and statistical definitions of probabilities, theorems of addition and multiplication of probabilities and their consequences, repetition of tests, discrete and continuous random variables, numerical characteristics of random variables. You will study the normal distribution in detail and at the end of the course you will get acquainted with some applications of the Monte Carlo method.
Each section of the course contains brief theoretical information and tasks for self-study.
Knowledge requirements: completion of the course Welcome to Engee.
Total course time: ~4 hours.
Course program
Basic concepts of probability theory
The types of random events, classical and statistical definitions of probabilities, properties of probability, basic formulas of combinatorics and calculation of probabilities with their help are studied.
Theorems of addition and multiplication of probabilities. The consequences of these
The theorem of probability addition, conditional probability, probability multiplication theorem, the concept of independent events, the probability of occurrence of at least one event, the formula of total probability and the Bayes formula are studied.
Repeating the tests
Bernoulli’s formula, Laplace’s local and integral theorems, and Poisson’s formula are studied.
Discrete random variables
The law of probability distribution of a discrete random variable, binomial distribution, Poisson distribution, the simplest flow of events, geometric distribution, numerical characteristics of discrete random variables (mathematical expectation, variance and mean square deviation) are studied.
Continuous random variables
The distribution function, distribution density and numerical characteristics of continuous random variables, uniform and exponential distributions are studied.
Normal distribution
The density of the normal distribution, the distribution function, the probability of a normal random variable falling into a given interval, the probability of a given deviation, the three sigma rule, the central limit theorem, skewness and kurtosis, and the chi-square distribution are studied.
The Monte Carlo method
Basic information is given about the essence of the Monte Carlo method, its errors, the generation of random numbers in Engee, examples of the application of the Monte Carlo method (calculation of the value of the number π, calculation of certain integrals, one-dimensional and two-dimensional random walks).