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Published in:
2022 | OriginalPaper | Chapter
7. Probability Theory
Author : Peter Müller
Published in: Handbook of Dynamics and Probability
Publisher: Springer International Publishing
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Abstract
This chapter gives the axiomatic approach to probability theory, due to Kolmogorov. It defines the probability space and then introduces the important notions of conditional probabilities and statistical independence. The central concept of a random variable and its probability distribution and its expectation is introduced next. Moments, characteristic functions, generating functions, and escort distributions are various ways to describe the probability distributions of random variables. Then families of two or a finite number of random variables are considered, so-called bi- and multivariate random variables. Finally, we consider two important theorems which describe the limiting behavior of mean values, the weak and strong law of large numbers. Limits of sequences of random variables can also be used to define probability distributions. One example is the central limit theorem which establishes the importance of the Gaussian distribution. Another example are the generalized extreme value distributions.