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[[File:Discrete probability distribution illustration.svgrightthumbFrom top to bottom, the cumulative distribution function of a discrete probability distribution, continuous probability distribution, and a distribution which has both a continuous part and a discrete part.]]
Every cumulative distribution function <math>F_X</math> is [[monotone increasingnondecreasing]]<ref name=KunIlPark/>{{rpp. 78}} and [[rightcontinuous]],<ref name=KunIlPark/>{{rpp. 79}}
:<math>\lim_{x\to \infty}F_X(x)=0, \quad \lim_{x\to +\infty}F_X(x)=1.</math>
Ztable:
One of the most popular application of cumulative distribution function is [[standard normal table]], also called the '''unit normal table''' or '''Z table''',<ref>{{Cite weburl=https://www.ztable.net/title=Z Tablelast=first=date=website=Z Tablelanguage=enUSaccessdate=20191211}}</ref>
;Properties
For two continuous variables ''X'' and ''Y'': <math> \Pr(a<X<b \text{ and } c<Y<d) = \int\limits_a^b \int\limits_c^d f(x,y) \, dy \, dx</math>;
For two discrete random variables, it is beneficial to generate a table of probabilities and address the cumulative probability for each potential range of ''X'' and ''Y'', and here is the example:<ref>{{Cite weburl=https://math.info/Probability/Joint_CDF/title=Joint Cumulative Distribution Function (CDF)website=math.infoaccessdate=20191211}}</ref>
given the joint probability density function in tabular form, determine the joint cumulative distribution function.
