Generating functions and their use in the theory of probability

Generating functions are suitable mathematical apparatus to describe the distribution of random variables. In this paper we introduce frequently used types of generating functions, their basic properties, uniqueness and benefits of use. We use this established mathematical apparatus for some known continuous and discrete distributions, but also the unusual practical examples. Then we fully utilize the strength of generating functions to solve interesting problems, often associated with creation of the theory of branching processes, an important area of probability theory