StatlectThe Digital Textbook

# Absolutely continuous random variable

An absolutely continuous random variable is a random variable whose cumulative distribution function is a continuous function. The set of values it can take is not countable and these values, when taken one by one, have zero probability of being observed.

## Synonyms

The adverb "absolutely" can be omitted. The term continuous random variable is a synonym of absolutely continuous random variable.

## Definition

The following is a formal definition.

Definition A random variable is said to be absolutely continuous if the probability that it assumes a value in a given interval can be expressed as an integral:where the integrand function is called the probability density function of .

Note that, as a consequence of this definition, the cumulative distribution function of isBecause integrals are continuous with respect to their upper bound of integration, is continuous in , which explains why we have stated above that a continuous random variable is a random variable whose cumulative distribution function is continuous.

## Example

Let be an absolutely continuous random variable that can take any value in the interval . Let its probability density function beThen, for example, the probability that takes a value between and can be computed as follows:

## More details

Absolutely continuous random variables are discussed in more detail in the lecture entitled Random variables. You can also read a brief introduction to the probability density function, including some examples, in the glossary entry entitled Probability density function.