Editing Probability distribution (section)

== Absolutely continuous probability distribution==
{{Main|Probability density function}}

An '''absolutely continuous probability distribution''' is a probability distribution on the real numbers with uncountably many possible values, such as a whole interval in the real line, and where the probability of any event can be expressed as an integral.<ref>{{Cite book|title=A First Look at Rigorous Probability Theory|author1=Jeffrey Seth Rosenthal|date=2000| publisher=World Scientific}}</ref> More precisely, a real random variable <math>X</math> has an [[absolutely continuous]] probability distribution if there is a function <math>f: \Reals \to [0, \infty]</math> such that for each interval <math>I = [a,b] \subset \mathbb{R}</math> the probability of <math>X</math> belonging to <math>I</math> is given by the integral of <math>f</math> over <math>I</math>:<ref>Chapter 3.2 of {{harvp|DeGroot|Schervish|2002}}</ref><ref>{{Cite web| last=Bourne|first=Murray|title=11. Probability Distributions - Concepts|url=https://www.intmath.com/counting-probability/11-probability-distributions-concepts.php|access-date=2020-09-10|website=www.intmath.com|language=en-us}}</ref>
<math display="block">P\left(a \le X \le b \right) = \int_a^b f(x) \, dx .</math>
This is the definition of a [[probability density function]], so that absolutely continuous probability distributions are exactly those with a probability density function.
In particular, the probability for <math>X</math> to take any single value <math>a</math> (that is, <math>a \le X \le a</math>) is zero, because an [[integral]] with coinciding upper and lower limits is always equal to zero.
If the interval <math>[a,b]</math> is replaced by any measurable set <math>A</math>, the according equality still holds:
<math display="block"> P(X \in A) = \int_A f(x) \, dx .</math>

An '''absolutely continuous random variable''' is a random variable whose probability distribution is absolutely continuous.

There are many examples of absolutely continuous probability distributions: [[normal distribution|normal]], [[Uniform distribution (continuous)|uniform]], [[Chi-squared distribution|chi-squared]], and [[List of probability distributions#Absolutely continuous distributions|others]].

=== 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]].