Editing Probability distribution (section)

=== Cumulative distribution function ===
Absolutely continuous probability distributions as defined above are precisely those with an [[Absolute continuity|absolutely continuous]] cumulative distribution function.
In this case, the cumulative distribution function <math>F</math> has the form
<math display="block">F(x) = P(X \leq x) = \int_{-\infty}^x f(t)\,dt</math>
where <math>f</math> is a density of the random variable <math>X</math> with regard to the distribution <math>P</math>.

''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 distribution]]s, 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 function|absolutely continuous]], i.e. refer to absolutely continuous distributions as continuous distributions.<ref name="ross">{{cite book|first=Sheldon M.|last=Ross|title=A first course in probability|publisher=Pearson|year=2010}}</ref> 

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