Throughout this chapter, a “population” means a population of numbers. When randomly selecting a number from the population, that constructs a random variable. We will denote random variables with an upper case letter, such as When referring to a specific value that can occur, we will use a lower case letter, such as The notation means “the event that a randomly generated is equal to ” Think of as representing all possible values, whereas is a particular value. We will use the convention for the remained of the text.
The makeup of the population determines the probabilities of witnessing values of the random variable. This means that a random variable always has as companion a function, where this function describes appropriately all probabilities of possible events for the random variable This “companion function” is called a distribution, and how it is defined and used is heavily impacted by whether the random variable is discrete or continuous. This is discussed in the next two sections.