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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 X is said to be absolutely continuous if the probability that it assumes a value in a given interval $\left[ a,b\right] $ can be expressed as an integral:[eq1]where the integrand function [eq2] is called the probability density function of X.

Note that, as a consequence of this definition, the cumulative distribution function of $X~$is[eq3]Because integrals are continuous with respect to their upper bound of integration, [eq4] is continuous in x, which explains why we have stated above that a continuous random variable is a random variable whose cumulative distribution function is continuous.

Example

Let X be an absolutely continuous random variable that can take any value in the interval $\left[ 0,1\right] $. Let its probability density function be[eq5]Then, for example, the probability that X takes a value between $1/2$ and 1 can be computed as follows:[eq6]

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.

Keep reading the glossary

Next entry: Absolutely continuous random vector

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