Abstract
The theory of distributions (generalized functions) is an effective tool in probability theory and the theory of random (stochastic) processes and fields. The aim of this chapter is to introduce the basic definitions and ideas of probability theory, and show how the simple distribution-theoretic concepts of the Dirac delta, its derivatives, and other related distributions can constructively be used to streamline statistical calculations. This approach is seldom used in the standard probability and statistics textbooks but is quite popular and effective in the physical and engineering sciences. In what follows we deliberately shall not resort to the rigorous mathematical measure-theoretic probability theory based on Kolmogorov’s axioms; such an approach is beyond the scope of this textbook. In order to develop the distributional tools discussed below, an intuitive “physical” idea of an ensemble of realizations of random quantities, processes, fields, and the concept of statistical averaging over the ensemble of realizations are sufficient.
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- 1.
There is, of course, a huge literature on the mathematical probability theory. We just quote here a few of the classics: Feller [1], Loève [2], Billingsley [3], and Kallenberg [4].
- 2.
For a more comprehensive discussion of different ways randomness can arise and be formally analyzed, see, M. Denker and W.A. Woyczyński [5], Introductory Statistics and Random Phenomena: Uncertainty, Complexity and Chaotic Behavior in Engineering and Science, Birkhauser-Boston 1998.
- 3.
For a more detailed discussion of random number generators and what is meant by their “randomness” see, e.g., the above-quoted textbook by Denker and Woyczyński.
- 4.
The picture was produced by my (WAW) student, Andrew Freedman, as part of a project in one of my stochastic processes classes.
- 5.
This example is discussed in greater detail in Keller [6], and Guttorp [7]. An interesting experiment by Persi Diaconis, to establish the likely practical values of v, and \(\omega \), is also described in [6].
- 6.
It is common usage to apply the term “distribution” to both, distributions (generalized functions) like the Dirac delta and the probability distributions. This is particularly unfortunate in a book like ours since both concepts appear simultaneously. So, we will make sure that the two are not confused by always including the adjective “probability” when discussing probability cumulative distribution.
- 7.
In later chapters, where we delve into more purely mathematical issues of probability theory, we will also adopt the traditional in mathematics terminology: The mean value of X will be called the expectation of X and denoted \(\mathbf{E}X\).
- 8.
Recall that the Heaviside function (or, unit step function) \(\chi (x)\) is defined as being equal to 0, for \(x<0\), and equal to 1, for \(x\ge 0.\)
- 9.
In this chapter we just restrict ourselves to the study of either discrete, or (absolutely) continuous distribution functions, when the formula is correct, and exclude the so-called singular distributions when the derivative of the distribution function need not integrate to 1. We will consider singular probability distributions in Part VI devoted to anomalous fractional dynamics.
- 10.
Pronounced, probability density of X at x, given that \( Y =y\), with the analogous phrasing for the second conditional density.
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Saichev, A.I., Woyczyński, W.A. (2018). Basic Distributional Tools for Probability Theory. In: Distributions in the Physical and Engineering Sciences, Volume 3. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-92586-8_16
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DOI: https://doi.org/10.1007/978-3-319-92586-8_16
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