## 4.0.0 Introduction

Remember that discrete random variables can take only a countable number of possible values. On the other
hand, a continuous random variable $X$ has a range in the form of an interval or a union of non-overlapping
intervals on the real line (possibly the whole real line). Also, for any $x \in \mathbb{R}$, $P(X=x)=0$.
Thus, we need to develop new tools to deal with continuous random variables. The good news is that the
theory of continuous random variables is completely analogous to the theory of discrete random variables.
Indeed, if we want to oversimplify things, we might say the following: take any formula about discrete
random variables, and then replace *sums* with *integrals*, and replace *PMFs* with
probability density functions (*PDFs*), and you will get the corresponding formula for continuous
random variables. Of course, there is a little bit more to the story and that's why we need a chapter
to discuss it. In this chapter, we will also introduce mixed random variables that are mixtures of discrete
and continuous random variables.