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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.

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


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.


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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