Handbook of probability
The present book provides the fundamental notions of probability theory, which are summarized in 14 chapters.NEWLINENEWLINEChapters 1 and 2 deal with the basic definitions of probability theory, such as probability spaces, sigma algebras, and probability measures. The basic properties of probability measures are presented and the definition of conditional probability is given.NEWLINENEWLINEChapters 3, 4, and 5 present a detailed study of random variables. After a general discussion of random variables in Chapter~3, discrete and continuous random variables are analysed in Chapters 4 and 5, respectively. The most popular discrete distributions, such as discrete uniform distribution, Bernoulli distribution, binomial distribution, geometric distribution, negative binomial distribution, hypergeometric distribution and Poisson distribution, are introduced and their properties are discussed in Chapter~4, while Chapter~5 presents the most commonly used families of the continuous distributions, such as uniform distribution, exponential distribution, normal distribution, gamma distribution, beta distribution, Student \(t\)-distribution, Pareto distribution, log-normal distribution, Laplace distribution, double exponential distribution.NEWLINENEWLINEIn Chapter 6, methods of generating random variables are discussed. This is done for both univariate and multivariate random variables. Chapter~7 deals with random vectors, where both the discrete and the continuous random vectors are treated.NEWLINENEWLINEThe characteristic functions and the moment-generating functions for random variables and vectors are introduced in Chapters 8 and 9, respectively. The expressions of the characteristic function are presented for most commonly used distributions. Also, the relationship between the moment-generating function and the characteristic function is discussed and the properties of both functions are provided.NEWLINENEWLINEChapter~10 describes the Gaussian random vectors that are extensively used in practice. The relationship between the independence and the uncorrelatedness is discussed. Also, Cochran's theorem is formulated.NEWLINENEWLINEChapters 11 and 12 describe various types of convergence for sequences of random variables. Their relationships are treated in Chapter~11 in detail. The most famous limit theorems, such as the law of large numbers and the central limit theorem, are discussed in Chapter~12.NEWLINENEWLINETwo appendices are given as Chapters 13 and 14 and focus on the more general integration theory and moments of random variables with any distribution.
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