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==Definition==
The cumulative distribution function of a realvalued [[random variable]] <math>X</math> is the function
{{Equation box 1
==Derived functions==
===Complementary cumulative distribution function (tail distribution)===<! This section is linked from [[Power law]], [[Stretched exponential function]] and [[Weibull distribution]] >
Sometimes, it is useful to study the opposite question and ask how often the random variable is ''above'' a particular level.
:<math>\bar F_X(x) = \operatorname{P}(X > x) = 1  F_X(x).</math>
This has applications in [[statisticsstatistical]] [[hypothesis test]]ing, for example, because the onesided [[pvalue]] is the probability of observing a test statistic ''at least'' as extreme as the one observed. Thus, provided that the [[test statistic]], ''T'', has a continuous distribution,
:<math>p= \operatorname{P}(T \ge t) = \operatorname{P}(T > t) =1  F_T(t).</math>
:Then, on recognizing <math>\bar F_X(c) = \int_c^\infty f_X(x) \, dx </math> and rearranging terms,
::<math>
0 \leq c\bar F_X(c) \leq \operatorname{E}(X)  \int_0^c x f_X(x) \, dx \to 0 \text{
</math>
:as claimed.
 volume = 81  issue = 8  pages = 1179–1182
 year = 2011
 pmid =
}}<</ref>) of the distribution or of the empirical results.
If the CDF ''F'' is strictly increasing and continuous then <math> F^{1}( p ), p \in [0,1], </math> is the unique real number <math> x </math> such that <math> F(x) = p </math>. In such a case, this defines the '''inverse distribution function''' or [[quantile function]].
Some distributions do not have a unique inverse (for example in the case where <math>f_X(x)=0</math> for all <math>a<x<b</math>, causing <math>F_X</math> to be constant).
:<math>
F^{1}(p) = \inf
</math>
* Example 1: The median is <math>F^{1}( 0.5 )</math>.
* Example 2: Put <math> \tau = F^{1}( 0.95 ) </math>.
Some useful properties of the inverse cdf (which are also preserved in the definition of the generalized inverse distribution function) are:
==Use in statistical analysis==
The concept of the cumulative distribution function makes an explicit appearance in statistical analysis in two (similar) ways. [[Cumulative frequency analysis]] is the analysis of the frequency of occurrence of values of a phenomenon less than a reference value. The [[empirical distribution function]] is a formal direct estimate of the cumulative distribution function for which simple statistical properties can be derived and which can form the basis of various [[statistical hypothesis test]]s. Such tests can assess whether there is evidence against a sample of data having arisen from a given distribution, or evidence against two samples of data having arisen from
===Kolmogorov–Smirnov and Kuiper's tests===

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