Cumulative distribution function: Difference between revisions

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==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. This is called the '''complementary cumulative distribution function''' ('''ccdf''') or simply the '''tail distribution''' or '''exceedance''', and is defined as
 
:<math>\bar F_X(x) = \operatorname{P}(X > x) = 1 - F_X(x).</math>
 
This has applications in [[statistics|statistical]] [[hypothesis test]]ing, for example, because the one-sided [[p-value]] 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, the one-sided [[p-value]] is simply given by the ccdf: for an observed value <math>t</math> of the test statistic
:<math>p= \operatorname{P}(T \ge t) = \operatorname{P}(T > t) =1 - F_T(t).</math>
 
In [[survival analysis]], <math>\bar F_X(x)</math> is called the '''[[survival function]]''' and denoted <math> S(x) </math>, while the term ''reliability function'' is common in [[engineering]].
 
;Properties
* For a non-negative continuous random variable having an expectation, [[Markov's inequality]] states that<ref name="ZK">{{cite book| last1 = Zwillinger| first1 = Daniel| last2 = Kokoska| first2 = Stephen| title = CRC Standard Probability and Statistics Tables and Formulae| year = 2010| publisher = CRC Press| isbn = 978-1-58488-059-2| page = 49 }}</ref>
:: <math>\bar F_X(x) \leq \frac{\operatorname{E}(X)}{x} .</math>
* As <math> x \to \infty, \bar F_X(x) \to 0 \ </math>, and in fact <math> \bar F_X(x) = o(1/x) </math> provided that <math>\operatorname{E}(X)</math> is finite.
:Proof:{{citation needed|date=April 2012}} Assuming <math>X</math> has a density function <math>f_X</math>, for any <math> c> 0 </math>
::<math>
\operatorname{E}(X) = \int_0^\infty x f_X(x) \, dx \geq \int_0^c x f_X(x) \, dx + c\int_c^\infty f_X(x) \, dx
</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{ as } c \to \infty
</math>
:as claimed.
 
===Folded cumulative distribution===
[[Image:Folded-cumulative-distribution-function.svg|thumb|right|Example of the folded cumulative distribution for a [[normal distribution]] function with an [[expected value]] of 0 and a [[standard deviation]] of 1.]]
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