This rather long chapter constitutes part of the hike in our walk/hike/stroll set-up. We introduce the reader to the basics of stochastics (representing both probability and statistics) necessary for the more technical discussions on risk later. The path followed starts from probability space (a theoretical concept we quickly leave aside); we then move to the notion of a random variable and,, its distribution function, including the most important discrete as well as continuous examples. Historical examples as well as pedagogical ones are always included in order to support the understanding of the new concepts introduced. These examples often show that there is more to randomness than meets the eye. For the applications discussed later, we will measure statistical uncertainty through the concept of confidence intervals. These can be based either on some asymptotic theory involving the famous bell curve, the normal distribution, or on some form of resampling known under the name of bootstrapping. Further, we add some tools that are very important for measuring and communicating risk; these include the concepts of return periods and quantile functions.

In this chapter we provide an overview of data modeling and describe the formulation of probabilistic models. We introduce random variables, their probability distributions, associated probability densities, examples of common densities, and the fundamental theorem of simulation to draw samples from discrete or continuous probability distributions. We then present the mathematical machinery required in describing and handling probabilistic models, including models with complex variable dependencies. In doing so, we introduce the concepts of joint, conditional, and marginal probability distributions, marginalization, and ancestral sampling.

Chapter 2 introduces the need for wireless network performance analysis tools to drive optimal network deployments and set optimal parameter values and describes the main building blocks and models of any wireless network performance analysis tool. In more details, it focuses on i) the system-level simulation and ii) the theoretical performance analysis concepts used in this book, paying particular attention to stochastic geometry frameworks.

Chapter 2 provides a review of probability theory, focusing on the topics that are essential for the remainder of the book. Included are the elements of set theory, the axioms and basic rules of probability theory, the concept of a random variable, discrete and continuous random variables, univariate and multivariate probability distributions, reliability and hazard functions, expectation and statistical moments, distributions and moments of functions of random variables, and extreme-value distributions. Appendix A presents commonly used probability distribution models with their properties, for easy reference. Thorough mastery of the material in this chapter is essential for understanding the remainder of the book.

We give a short introduction to some classical information-theoretic quantities: joint entropy, conditional entropy and mutual information. We then interpret their exponentials ecologically, as meaningful measures of subcommunities of a larger metacommunity. These subcommunity and metacommunity measures have excellent logical properties, as we establish. We also show how all these quantities can be presented in terms of relative entropy and the value measures of the previous chapter.

We continue the study initiated in Chapter 7 of polynomials with small norms. This time the norm of the polynomial is not taken as the supremum on the n-dimensional polydisc, we take it on B_X, the unit ball of some Banach space. The goal is to show that, given a polynomial, signs can be found in such a way that the norm of the new polynomial, whose coefficients are the original ones multiplied by the signs, has small norm. We do this with three different approaches. The first two approaches use Rademacher random variables as the main probabilistic tools. The first one is based on finding out how many balls of a fixed radius are needed to cover B_X while the second one uses entropy integrals and a good estimate for the entropy numbers of the inclusions between l_p spaces. The third approach is different, and relies on Gaussian random variables, Slepian’s lemma and the fact that Rademacher averages are dominated by Gaussian averages. This approach also allows to get estimates for vector-valued polynomials.