Abstract
In this chapter, after recapitulating the elements of probability theory, we introduce the notion of random variables and probability distribution of random variables. Next, we discuss in turn joint, marginal, and conditional distribution of random variables, as well as such attributes of a probability distribution of a random variable as mean, variance, standard deviation, raw, and central moments. We also discuss the concepts of covariance and correlation between random variables and that of moment and cumulant generating functions of a random variable. In the following section, we provide some familiar examples of probability distributions and then move on to discuss two central results related to sums of independent and identically-distributed (i.i.d.) random variables, the Law of Large Numbers, and the Central Limit Theorem. We then discuss stable distributions, which allow a generalization of the Central Limit theorem. This is followed by a brief overview of the elements of the extreme value theory that deals with extreme values of a sample of i.i.d. random variables. As an illustration of the concepts developed, we discuss in detail the paradigmatic example of a random walk in one dimension.
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Das, D., Gupta, S. (2023). Random Variables and Probability Distributions. In: Facets of Noise. Fundamental Theories of Physics, vol 214. Springer, Cham. https://doi.org/10.1007/978-3-031-45312-0_1
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DOI: https://doi.org/10.1007/978-3-031-45312-0_1
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