Measurements are often numerical in nature, which naturally leads to distributions on the real line. We start our discussion of such distributions in the present chapter, and in the process introduce the concept of random variable, which is really a device to facilitate the writing of probability statements and the derivation of the corresponding computations. We introduce objects such as the distribution function, survival function, and quantile function, any of which characterizes in the underlying distribution.

Some experiments lead to considering not one, but several measurements. As before, each measurement is represented by a random variable, and these are stacked into a random vector. For example, in the context of an experiment that consists in flipping a coin multiple times, we defined in a previous chapter as many random variables, each indicating the result of one coin flip. These are then concatenated to form a random vector, compactly describing the outcome of the entire experiment. Concepts such as conditional probability and independence are introduced.

We consider an experiment that yields, as data, a sample of independent and identically distributed (real-valued) random variables with a common distribution on the real line. The estimation of the underlying mean and median is discussed at length, and bootstrap confidence intervals are constructed. Tests comparing the underlying distribution to a given distribution (e.g., the standard normal distribution) or a family of distribution (e.g., the normal family of distributions) are introduced. Censoring, which is very common in some clinical trials, is briefly discuss.

In this chapter we introduce some tools for sampling from a distribution. We also explain how to use computer simulations to approximate probabilities and, more generally, expectations, which can allow one to circumvent complicated mathematical derivations. The methods that are introduced include Monte Carlo sampling/integration, rejection sampling, and Markov Chain Monte Carlo sampling.

An expectation is simply a weighted mean, and means are at the core of Probability Theory and Statistics. In Statistics, in particular, such expectations are used to define parameters of interest. It turns out that an expectation can be approximated by an empirical average based on a sample from the distribution of interest, and the accuracy of this approximation can be quantified via what is referred to as concentration inequalities.

An empirical average will converge, in some sense, to the corresponding expectation. This famous result, called the Law of Large Numbers, can be anticipated based on the concentration inequalities introduced in the previous chapter, but some appropriate notions of convergence for random variables need to be defined in order to make a rigorous statement. Beyond mere convergence, the fluctuations of an empirical average around the associated expectation can be characterized by the Central Limit Theorem, and are known to be Gaussian in some asymptotic sense. The chapter also discusses the limit of extremes such as the maximum of a sample.

Stochastic processes model experiments whose outcomes are collections of variables organized in some fashion. We focus here on Markov processes, which include random walks (think of the fortune of a person gambling on black/red at the roulette over time) and branching processes (think of the behavior of a population of an asexual species where each individual gives birth to a number of otherwise identical offsprings according to a given probability distribution) .

In this chapter we consider distributions on the real line that have a discrete support. It is indeed common to count certain occurrences in an experiment, and the corresponding counts are invariably integer-valued. In fact, all the major distributions of this type are supported on the (non-negative) integers. We introduce the main ones here.

We consider an experiment resulting in two paired numerical variables. The general goal addressed in this chapter is that of quantifying the strength of association between these two variables. By association we mean dependence. Contrary to the previous chapter, here the two variables can be measurements of completely different kinds (e.g., height and weight). Several measures of association are introduced, and used to test for independence.

In some areas of mathematics, physics, and elsewhere, continuous objects and structures are often motivated, or even defined, as limits of discrete objects. For example, in mathematics, the real numbers are defined as the limit of sequences of rational numbers, and in physics, the laws of thermodynamics arise as the number of particles in a system tends to infinity (the so-called thermodynamic or macroscopic limit). Taking certain discrete distributions (discussed in the previous chapter) to their continuous limits, which is done by letting their support size increase to infinity in a controlled manner, gives rise to continuous distributions on the real line. We introduce and discuss such distributions in this chapter, including the normal (aka Gaussian) family of distributions, and in the process cover probability densities.

We consider in this chapter experiments where the variables of interest are paired. Importantly, we assume that these variables are directly comparable (in contrast with the following two chapters). Crossover trials are important examples of such experiments. The main question of interest here is that of exchangeability, which reduces to testing for symmetry when there are only two variables.

When a die (with 3 or more faces) is rolled, the result of each trial can take one of as many possible values. The same is true in the context of an urn experiment, when the balls in the urn are of multiple different colors. Such models are broadly applicable. Indeed, even `yes/no’ polls almost always include at least one other option like `not sure’ or `no opinion’. Another situation where discrete variables arise is when two or more coins are compared in terms of their chances of landing heads, or more generally, when two or more (otherwise identical) dice are compared in terms of their chances of landing on a particular face. In terms of urn experiments, the analog is a situation where balls are drawn from multiple urns. This sort of experiments can be used to model clinical trials where several treatments are compared and the outcome is dichotomous. When the coins are tossed together, or when the dice are rolled together, we might want to test for independence. We thus introduce some classical tests for comparing multiple discrete distributions and for testing for the independence of two or more discrete variables that are observed together.