Cumulative distribution function: Difference between revisions

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Z-table:
 
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=https://www.ztable.net/|title=Z Table|last=|first=|date=|website=Z Table|language=en-US|url-status=live|archive-url=|archive-date=|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.
 
;Properties
# <math>\lim_{x_1,\ldots,x_n \rightarrow+\infty}F_{X_1 \ldots X_n}(x_1,\ldots,x_n)=1 \text{ and } \lim_{x_i\rightarrow-\infty}F_{X_1 \ldots X_n}(x_1,\ldots,x_n)=0, \text{for all } i.</math>
 
The probability that a point belongs to a [[hyperrectangle]] is analogous to the 1-dimensional case:<ref>[https://web.archive.org/web/20160222051842/http://www.math.wustl.edu/~hgan/Prob2014/slides.259-327.pdf]</ref>
:<math>F_{X_1,X_2}(a, c) + F_{X_1,X_2}(b, d) - F_{X_1,X_2}(a, d) - F_{X_1,X_2}(b, c) = \operatorname{P}(a < X_1 \leq b, c < X_2 \leq d) = \int ...</math>
 
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