There are experiments that produce a collection of n ≥ 2 random variables. For most of this chapter, the focus is on two random variables (n = 2) and their relationships, as the study of a pair of random variables is sufficient to introduce all relevant concepts and important techniques to calculate the probability of interest for any number of random variables. A discussion of random vectors for n > 2 is also briefly provided.
6.1 Pairs of Random Variables
In a random experiment, there may be a need to measure two quantities of interest, and the probability of interest may involve both random quantities. Examples include the transmitted and received signals in a communication system, heights and weights of newborn babies, education level and level of income, cost and reliability, cigarette smoking and lung cancer, texting and car accidents, tax cut and social assistance, and alcohol consumption and drug use, to name just a few. There are many applications of interest in a multitude of fields in science and engineering that can be handled by the theory of two random variables. In practice, quite often, we observe one random variable, but we want to know about the other random variable that is inaccessible.
In our notation, the names of the two random variables are always uppercase letters, such as X and Y, and lowercase letters denote possible values of the two random variables, such as x and y. In an experiment, in which there are two random variables ...