Editing Probability distribution (section)

==Cumulative distribution function==
In the special case of a real-valued random variable, the probability distribution can equivalently be represented by a cumulative distribution function instead of a probability measure. The cumulative distribution function of a random variable <math>X</math> with regard to a probability distribution <math>p</math> is defined as
<math display="block">F(x) = P(X \leq x).</math>

The cumulative distribution function of any real-valued random variable has the properties:
*<li style="margin: 0.7rem 0;"><math>F(x)</math> is non-decreasing;</li>
*<li style="margin: 0.7rem 0;"><math>F(x)</math> is [[right-continuous]];</li>
*<li style="margin: 0.7rem 0;"><math>0 \le F(x) \le 1</math>;</li>
*<li style="margin: 0.7rem 0;"><math>\lim_{x \to -\infty} F(x) = 0</math> and <math>\lim_{x \to \infty} F(x) = 1</math>; and</li>
*<li style="margin: 0.7rem 0;"><math>\Pr(a < X \le b) = F(b) - F(a)</math>.</li>

Conversely, any function <math>F:\mathbb{R}\to\mathbb{R}</math> that satisfies the first four of the properties above is the cumulative distribution function of some probability distribution on the real numbers.<ref>{{Cite book|title=Probability and stochastics|last=Erhan|first=Çınlar|date=2011|publisher=Springer|isbn=9780387878584|location=New York|pages=57}}</ref>

Any probability distribution can be decomposed as the [[mixture distribution|mixture]] of a [[Discrete probability distribution|discrete]], an [[Absolutely continuous probability distribution|absolutely continuous]] and a [[Singular distribution|singular continuous distribution]],<ref>see [[Lebesgue's decomposition theorem]]</ref> and thus any cumulative distribution function admits a decomposition as the [[convex sum]] of the three according cumulative distribution functions.