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Cumulative distribution function

Absolutely continuous probability distributions as defined above are precisely those with an absolutely continuous cumulative distribution function.In this case, the cumulative distribution function F {\displaystyle F} has the form F (x) = P (X ≤ x) = ∫ − ∞ x f (t) d t {\displaystyle F(x)=P(X\leq x)=\int _{-\infty }^{x}f(t)\,dt} where f {\displaystyle f} is a density of the random variable X {\displaystyle X} with regard to the distribution P {\displaystyle P}.

Note on terminology: Absolutely continuous distributions ought to be distinguished from continuous distributions, which are those having a continuous cumulative distribution function. Every absolutely continuous distribution is a continuous distribution but the inverse is not true, there exist singular distributions, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the Cantor distribution. Some authors however use the term "continuous distribution" to denote all distributions whose cumulative distribution function is absolutely continuous, i.e. refer to absolutely continuous distributions as continuous distributions.

For a more general definition of density functions and the equivalent absolutely continuous measures see absolutely continuous measure.