This chapter introduces random variables and explains how to use them to model uncertain numerical quantities that are discrete. We first provide a mathematical definition of random variables, building upon the framework of probability spaces. Then, we explain how to manipulate discrete random variables in practice, using their probability mass function (pmf), and describe the main properties of the pmf. Motivated by an example where we analyze Kevin Durant's free-throw shooting, we define the empirical pmf, a nonparametric estimator of the pmf that does not make strong assumptions about the data. Next, we define several popular discrete parametric distributions (Bernoulli, binomial, geometric, and Poisson), which yield parametric estimators of the pmf, and explain how to fit them to data via maximum-likelihood estimation. We conclude the chapter by comparing the advantages and disadvantages of nonparametric and parametric models, illustrated by a real-data example, where we model the number of calls arriving at a call center.

We introduce the concepts of a map, a sigma-field generated by a map and the measurability of a map. This leads to the notion of random variable on a probability space, which is just a map being measurable with respect to the considered sigma-field. This guarantees that the distribution of a random variable is well-defined on a given probability space. Next, we review the main families of random variables, discrete (uniform, Bernoulli, binomial) and continuous (uniform, exponential, normal, log-normal), and recall their mass and density functions, as well as their cumulative distribution functions. In particular, we highlight that any random variable can be built by transforming a continuous uniform random variable in an appropriate manner, following the probability integral transform. Finally, we introduce random vectors (vector of random variables), joint and marginal distributions, and the independence property. We illustrate those concepts on toy examples as well as on our stock price model, computing the distribution of prices at various points in time. We explain how correlation can significantly impact the risk of a portfolio of stocks in simple discrete models.