Continuous Random Variables

Abstract

We mentioned in Chapter 4 that discrete random variables serve as good examples to develop probabilistic intuition, but they do not account for all the random variables that one studies in theory and applications. In this chapter, we introduce the so-called continuous random variables, which typically take all values in some nonempty interval; e.g., the unit interval, the entire real line, etc. The right probabilistic paradigm for continuous variables cannot be pmfs. Discrete probability, which is based on summing things, is replaced by integration when we deal with continuous random variables and, instead of pmfs, we operate with a density function for the variable The density function fully describes the distribution, and calculus occupies the place of discrete operations, such as sums, when we come to continuous random variables. The basic concepts and examples that illustrate how to do the basic calculations are discussed in this chapter. Distinguished continuous distributions that arise frequently in applications will be treated separately in later chapters.