Cumulative distribution function: Difference between revisions

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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|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.
# <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>[]</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>