Cumulative distribution function: Difference between revisions

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[[File:Discrete probability distribution illustration.svg|right|thumb|From 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 increasing|non-decreasing]]<ref name=KunIlPark/>{{rp|p. 78}} and [[right-continuous]],<ref name=KunIlPark/>{{rp|p. 79}}, which makes it a [[càdlàg]] function. Furthermore,
:<math>\lim_{x\to -\infty}F_X(x)=0, \quad \lim_{x\to +\infty}F_X(x)=1.</math>
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 web|url=|title=Z Table|last=|first=|date=|website=Z Table|language=en-US|access-date=2019-12-11}}</ref>, is the value of cumulative distribution function of the normal distribution. It is very useful to use Z-table not only for probabilities below a value which is the original application of cumulative distribution function, but also above and/or between values on standard normal distribution, and it was further extended to any normal distribution.
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 web|url=|title=Joint Cumulative Distribution Function (CDF)||access-date=2019-12-11}}</ref>:
given the joint probability density function in tabular form, determine the joint cumulative distribution function.