In Definitions for Discrete Random Variables, we mainly focused on discrete random variables whose set of possible values is either finite or countably infinite. In this chapter, we study random variables whose set of possible values is uncountable. We’ll see later on that a lot of the cases we’ve discussed have analogs in the continuous case.

### Probability density function

The continuous random variable is a random variable with infinite possible outcomes (a subset of the real line). We say is a

**continuous random variable**if there exists a nonnegative function defined for all , having the property that for any set of real numbers,Here the function is called the

**probability density function (PDF)**. It resembles the probability mass function in the discrete case. The PDF has the following properties:- .

- .

- .

We can also define the

**cumulative distribution function (CDF)**for a continuous random variable:#### Properties example

These properties often come in handy when we have unknown quantities in a PDF. Suppose is a continuous random variable with probability density function

and we’d like to find as well as the probability .

Now we can easily find . With the PDF given, it’s trivial to find the CDF:

#### Lifetime example

Suppose , the lifetime of an item, is a continuous random variable with a density function

What is the probability that the item functions between 50 and 150 days?

In mathematical terms, we want to calculate . We first need to find the value of .

Recall that , and because the derivative is a linear function.

With , we can calculate

### Expectation and variance

Earlier we’ve defined the expectation for discrete random variables. If is a continuous random variable with probability density function , we have

so it’s easy to find the analog for the

**expectation**of to beSimilarly, the

**expected value**of a real-valued function of iswhich can be used to derive the

**variance**of :Suppose is a continuous random variable with density function

Next, we introduce some Common Distributions of Continuous Random Variables.