# Science:MATH105 Probability/Lesson 2 CRV/2.04 Continuous Random Variables

The distinction between continuous and random variables can now be made more precise by using the CDF.

Definition: Continuous Random Variable |
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Let |

It is *by definition* that continuous CDFs of a random variable are CDFs of *continuous* random variables.

In our prior grade distribution example, the CDF was not continuous and so it corresponded to a discrete random variable that represented a grade. Whereas our previous temperature example, the CDF was continuous, the random variable was a continuous random variable that represented a temperature.

## Example

Consider the function whose graph is given below.

This function cannot represent a cumulative distribution function for a *continuous* random variable because *F* is not continuous for all values of *x*. However, *F* could represent a cumulative distribution function, because the limit as *x* goes to negative and positive infinity are 0 and 1, respectively.