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{{short description|probability that random variable X is less than or equal to x.}}
{{refimprove|date=March 2010}}

[[File:Exponential distribution cdf.png|thumb|300px|Cumulative distribution function for the [[Exponential distribution]]]]
[[File:Normal Distribution CDF.svg|thumb|300px|Cumulative distribution function for the [[Normal distribution]]]]

In [[probability theory]] and [[statistics]], the '''cumulative distribution function''' ('''CDF''') of a real-valued [[random variable]] <math>X</math>, or just '''distribution function''' of <math>X</math>, evaluated at <math>x</math>, is the [[probability]] that <math>X</math> will take a value less than or equal to <math>x</math><ref>{{Cite book|url=https://github.com/mml-book/mml-book.github.io|title=Mathematics for Machine Learning|last=Deisenroth,Faisal,Ong|first=Marc Peter,A Aldo, Cheng Soon|publisher=Cambridge University Press|year=2019|isbn=9781108455145|location=|pages=181}}</ref>. 

In the case of a scalar [[continuous distribution]], it gives the area under the [[probability density function]] from minus infinity to <math>x</math>. Cumulative distribution functions are also used to specify the distribution of [[multivariate random variable]]s.

==Definition==
The cumulative distribution function of a real-valued [[random variable]] <math>X</math> is the function given by<ref name=KunIlPark>{{cite book | author=Park, Kun Il| title=Fundamentals of Probability and Stochastic Processes with Applications to Communications| publisher=Springer | year=2018 | isbn=978-3-319-68074-3}}</ref>{{rp|p. 77}}

{{Equation box 1
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|equation = {{NumBlk||<math>F_X(x) = \operatorname{P}(X\leq x)</math>|{{EquationRef|Eq.1}}}}
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where the right-hand side represents the [[probability]] that the random variable <math>X</math> takes on a value less than or
equal to <math>x</math>. The probability that <math>X</math> lies in the semi-closed [[interval (mathematics)|interval]] <math>(a,b]</math>, where <math>a < b</math>, is therefore<ref name=KunIlPark/>{{rp|p. 84}}

{{Equation box 1
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|equation = {{NumBlk||<math>\operatorname{P}(a < X \le b)= F_X(b)-F_X(a)</math>|{{EquationRef|Eq.2}}}}
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In the definition above, the "less than or equal to" sign, "≤", is a convention, not a universally used one (e.g. Hungarian literature uses "<"), but is important for discrete distributions. The proper use of tables of the [[Binomial distribution|binomial]] and [[Poisson distribution]]s depends upon this convention. Moreover, important formulas like [[Paul Lévy (mathematician)|Paul Lévy]]'s inversion formula for the [[Characteristic function (probability theory)#Inversion formulae|characteristic function]] also rely on the "less than or equal" formulation.

If treating several random variables <math>X,Y,\ldots</math> etc. the corresponding letters are used as subscripts while, if treating only one, the subscript is usually omitted. It is conventional to use a capital <math>F</math> for a cumulative distribution function, in contrast to the lower-case <math>f</math> used for [[probability density function]]s and [[probability mass function]]s. This applies when discussing general distributions: some specific distributions have their own conventional notation, for example the [[normal distribution]].

The CDF of a [[continuous random variable]] <math>X</math> can be expressed as the integral of its probability density function <math>f_X</math> as follows:<ref name=KunIlPark/>{{rp|p. 86}}

:<math>F_X(x) = \int_{-\infty}^x f_X(t)\,dt.</math>

In the case of a random variable <math>X</math> which has distribution having a discrete component at a value <math>b</math>,
:<math>\operatorname{P}(X=b) = F_X(b) - \lim_{x \to b^{-}} F_X(x).</math>

If <math>F_X</math> is continuous at <math>b</math>, this equals zero and there is no discrete component at <math>b</math>.

== Properties ==
[[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>

Every function with these four properties is a CDF, i.e., for every such function, a [[random variable]] can be defined such that the function is the cumulative distribution function of that random variable.

If <math>X</math> is a purely [[discrete random variable]], then it attains values <math>x_1,x_2,\ldots</math> with probability <math>p_i = \operatorname{P}(x_i)</math>, and the CDF of <math>X</math> will be [[discontinuity (mathematics)|discontinuous]] at the points <math>x_i</math>:

:<math>F_X(x) = \operatorname{P}(X\leq x) = \sum_{x_i \leq x} \operatorname{P}(X = x_i) = \sum_{x_i \leq x} p(x_i).</math>

If the CDF <math>F_X</math> of a real valued random variable <math>X</math> is [[continuous function|continuous]], then <math>X</math> is a [[continuous random variable]]; if furthermore <math>F_X</math> is [[absolute continuity|absolutely continuous]], then there exists a [[Lebesgue integral|Lebesgue-integrable]] function <math>f_X(x)</math> such that

:<math>F_X(b)-F_X(a) = \operatorname{P}(a< X\leq b) = \int_a^b f_X(x)\,dx</math>

for all real numbers <math>a</math> and <math>b</math>. The function <math>f_X</math> is equal to the [[derivative]] of <math>F_X</math> [[almost everywhere]], and it is called the [[probability density function]] of the distribution of <math>X</math>.

== Examples ==
As an example, suppose <math>X</math> is [[uniform distribution (continuous)|uniformly distributed]] on the unit interval <math>[0,1]</math>.
Then the CDF of <math>X</math> is given by

:<math>F_X(x) = \begin{cases}
0 &:\ x < 0\\
x &:\ 0 \le x \le 1\\
1 &:\ x > 1
\end{cases}</math>

Suppose instead that <math>X</math> takes only the discrete values 0 and 1, with equal probability.
Then the CDF of <math>X</math> is given by

:<math>F_X(x) = \begin{cases}
0 &:\ x < 0\\
1/2 &:\ 0 \le x < 1\\
1 &:\ x \ge 1
\end{cases}</math>

==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.]]
While the plot of a cumulative distribution often has an S-like shape, an alternative illustration is the '''folded cumulative distribution''' or '''mountain plot''', which folds the top half of the graph over,<ref name="Gentle">{{cite book| author = Gentle, J.E.| title = Computational Statistics| url = https://books.google.com/?id=m4r-KVxpLsAC&pg=PA348| accessdate = 2010-08-06| year = 2009| publisher = [[Springer Science+Business Media|Springer]]| isbn = 978-0-387-98145-1 }}{{Page needed|date=June 2011}}</ref><ref name="Monti">
{{cite journal|author=Monti, K.L.|pages=342–345|year=1995|title=Folded Empirical Distribution Function Curves (Mountain Plots) |journal=The American Statistician|volume=49|issue=4|jstor=2684570|doi=10.2307/2684570}}</ref>
thus using two scales, one for the upslope and another for the downslope. This form of illustration emphasises the [[median (statistics)|median]] and [[dispersion (statistics)|dispersion]] (specifically, the [[mean absolute deviation]] from the median<ref>{{Cite journal
| last1 = Xue | first1 = J. H.
| last2 = Titterington | first2 = D. M.
| doi = 10.1016/j.spl.2011.03.014
| title = The p-folded cumulative distribution function and the mean absolute deviation from the p-quantile
| journal = Statistics & Probability Letters
| volume = 81 | issue = 8 | pages = 1179–1182
| year = 2011
| pmid = | pmc = | url = https://hal.archives-ouvertes.fr/hal-00753950/file/PEER_stage2_10.1016%252Fj.spl.2011.03.014.pdf
}}<</ref>) of the distribution or of the empirical results.

===Inverse distribution function (quantile function)===
{{main|Quantile function}}
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). This problem can be solved by defining, for <math> p \in [0,1] </math>, the '''generalized inverse distribution function''':
:<math>
F^{-1}(p) = \inf \{x \in \mathbb{R}: F(x) \geq p \}.
</math>
* Example 1: The median is <math>F^{-1}( 0.5 )</math>.
* Example 2: Put <math> \tau = F^{-1}( 0.95 ) </math>. Then we call <math> \tau </math> the 95th percentile.

Some useful properties of the inverse cdf (which are also preserved in the definition of the generalized inverse distribution function) are:

# <math>F^{-1}</math> is nondecreasing
# <math>F^{-1}(F(x)) \leq x</math>
# <math>F(F^{-1}(p)) \geq p</math>
# <math>F^{-1}(p) \leq x</math> if and only if <math>p \leq F(x)</math>
# If <math>Y</math> has a <math>U[0, 1]</math> distribution then <math>F^{-1}(Y)</math> is distributed as <math>F</math>. This is used in [[random number generation]] using the [[inverse transform sampling]]-method.
# If <math>\{X_\alpha\}</math> is a collection of independent <math>F</math>-distributed random variables defined on the same sample space, then there exist random variables <math>Y_\alpha</math> such that <math>Y_\alpha</math> is distributed as <math>U[0,1]</math> and <math>F^{-1}(Y_\alpha) = X_\alpha</math> with probability 1 for all <math>\alpha</math>.
The inverse of the cdf can be used to translate results obtained for the uniform distribution to other distributions.

==Multivariate case==
===Definition for two random variables===
When dealing simultaneously with more than one random variable the '''joint cumulative distribution function''' can also be defined. For example, for a pair of random variables <math>X,Y</math>, the joint CDF <math>F_{XY}</math> is given by<ref name=KunIlPark/>{{rp|p. 89}}

{{Equation box 1
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|equation = {{NumBlk||<math>F_{XY}(x,y) = \operatorname{P}(X\leq x,Y\leq y)</math>|{{EquationRef|Eq.3}}}}
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where the right-hand side represents the [[probability]] that the random variable <math>X</math> takes on a value less than or
equal to <math>x</math> '''and''' that <math>Y</math> takes on a value less than or
equal to <math>y</math>.

===Definition for more than two random variables===
For <math>N</math> random variables <math>X_1,\ldots,X_N</math>, the joint CDF <math>F_{X_1,\ldots,X_N}</math> is given by

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||<math>F_{X_1,\ldots,X_N}(x_1,\ldots,x_N) = \operatorname{P}(X_1 \leq x_1,\ldots,X_N \leq x_n)</math>|{{EquationRef|Eq.4}}}}
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Interpreting the <math>N</math> random variables as a [[random vector]] <math>\mathbf{X} = (X_1,\ldots,X_N)^T</math> yields a shorter notation:

:<math>F_{\mathbf{X}}(\mathbf{x}) = \operatorname{P}(X_1 \leq x_1,\ldots,X_N \leq x_n)</math>

===Properties===
Every multivariate CDF is:
# Monotonically non-decreasing for each of its variables,
# Right-continuous in each of its variables,
# <math>0\leq F_{X_1 \ldots X_n}(x_1,\ldots,x_n)\leq 1,</math>
# <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>

==Complex case==
===Complex random variable===
The generalization of the cumulative distribution function from real to [[complex random variable#Cumulative distribution function|complex random variables]] is not obvious because expressions of the form <math> P(Z \leq 1+2i) </math> make no sense. However expressions of the form <math> P(\Re{(Z)} \leq 1, \Im{(Z)} \leq 3) </math> make sense. Therefore, we define the cumulative distribution of a complex random variables via the [[Joint probability distribution|joint distribution]] of their real and imaginary parts:
:<math> F_Z(z)=F_{\Re{(Z)},\Im{(Z)}}(\Re{(z)},\Im{(z)})=P(\Re{(Z)} \leq \Re{(z)} , \Im{(Z)} \leq \Im{(z)}) </math>.

===Complex random vector===
Generalization of {{EquationNote|Eq.4}} yields
:<math>F_{\mathbf{Z}}(\mathbf{z}) = F_{\Re{(Z_1)},\Im{(Z_1)}, \ldots, \Re{(Z_n)},\Im{(Z_n)}}(\Re{(z_1)}, \Im{(z_1)},\ldots,\Re{(z_n)}, \Im{(z_n)}) = \operatorname{P}(\Re{(Z_1)} \leq \Re{(z_1)},\Im{(Z_1)} \leq \Im{(z_1)},\ldots,\Re{(Z_n)} \leq \Re{(z_n)},\Im{(Z_n)} \leq \Im{(z_n)})</math>
as definition for the CDS of a complex random vector <math>\mathbf{Z} = (Z_1,\ldots,Z_N)^T</math>.

==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 the same (unknown) population distribution.

===Kolmogorov–Smirnov and Kuiper's tests===
The [[Kolmogorov–Smirnov test]] is based on cumulative distribution functions and can be used to test to see whether two empirical distributions are different or whether an empirical distribution is different from an ideal distribution. The closely related [[Kuiper's test]] is useful if the domain of the distribution is cyclic as in day of the week. For instance Kuiper's test might be used to see if the number of tornadoes varies during the year or if sales of a product vary by day of the week or day of the month.

==See also==
* [[Descriptive statistics]]
* [[Distribution fitting]]
* [[Ogive (statistics)]]

== References ==
{{Reflist}}

== External links ==
* {{commons category-inline|Cumulative distribution functions}}

{{Theory of probability distributions}}

{{DEFAULTSORT:Cumulative Distribution Function}}
[[Category:Functions related to probability distributions]]