A **random variable**, usually denoted as X, is a variable whose values are numerical outcomes of some random process. There are two types of random variables: discrete and continuous. This section will focus on continuous random variables.

**Continuous Random Variables**

A **continuous random variable** is one which can take on an infinite number of possible values. Some examples of continuous random variables include:

Height of a person

Weight of an animal

Time required to run a mile

Amount of rainfall on a certain day

A continuous random variable is not defined at specific values. Instead, it’s defined over an interval of values and is represented by the area under a curve called a **probability density function**.

For example, suppose a restaurant advertises a burger that weighs a quarter-pound (0.25 lbs). Since *weight *is a continuous variable, it can take on an infinite number of values. For example, a given burger might actually weight 0.250001 pounds, or 0.24 pounds, or 0.2488 pounds. The probability that a given burger weights exactly .25 pounds is essentially zero.

Imagine that we record the weight of 100 random burgers from this restaurant and make a probability density histogram of the resulting weights:

Most of the burgers do weigh close to 0.25 pounds, but some weigh a little more or a little less. Now if we make the intervals even smaller on the x-axis, the density histogram might look like this:

If we keep making the intervals on the x-axis smaller and smaller, eventually the bars would become so narrow that we could connect the dots at the top of each bar and create a “curve” instead:

This curve represents the **probability density function** of our random variable *weight*.

**More on Probability Density Functions**

The total area under the curve is equal to one.

We can use a probability density function to find the probability that a random variable takes on a value within an interval, but we cannot use it to find the probability that a random variable takes on a specific value. For example, we can use the curve above to find the probability that a burger weighs between 0.23 and 0.27 pounds, but we cannot use it to find the probability that a burger weights exactly 0.25 pounds.

The most common continuous probability distribution is the normal distribution, which is described in detail in the next section.