# 7Continuous Random Variables

## 7.1 Introduction

With discrete random variables, we were often able to establish an equation to find the probability of specific values of the random variable. These equations had to be such that the sum of the probabilities of all possible values of the discrete random variable was 1. Special cases of distributions for discrete random variables such as the Binomial and the Poisson were discussed.

In this chapter, some properties of^{1}**continuous random variables** are introduced. There are many situations in real life where continuous random variables are needed. For example, when measuring something, like the weight of a product being manufactured or the operation time of a machine to complete a procedure, continuous random variables are defined. With a continuous random variable, we still need to work with its expected value, variance (or standard deviation), and probabilities. But these traits are calculated differently than with discrete random variables. Specifically, continuous random variables have an infinite amount of possible values that cannot be written down in an orderly fashion. The alternative approach is using a **probability density function**, , or pdf for short. A pdf is a function, which defines a curve such that the following two conditions hold:

- for all (it is a nonnegative function).
- The area under the entire curve must be 1.

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