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Cumulative distribution function

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Probability that random variable X is less than or equal to x

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Cumulative distribution function for theexponential distribution

Cumulative distribution function for thenormal distribution

Inprobability theory andstatistics, thecumulative distribution function (CDF) of a real-valuedrandom variable $X {\displaystyle X}$ , or justdistribution function of $X {\displaystyle X}$ , evaluated at $x {\displaystyle x}$ , is theprobability that $X {\displaystyle X}$ will take a value less than or equal to $x {\displaystyle x}$ .^[1]

Everyprobability distribution supported on the real numbers, discrete or "mixed" as well ascontinuous, is uniquely identified by aright-continuous monotone increasing function (acàdlàg function) $F\colon \mathbb {R} \rightarrow [0,1]$ satisfying $\lim _{x\rightarrow -\infty }F(x)=0$ and $\lim _{x\rightarrow \infty }F(x)=1$ .

In the case of a scalarcontinuous distribution, it gives the area under theprobability density function from negative infinity to $x {\displaystyle x}$ . Cumulative distribution functions are also used to specify the distribution ofmultivariate random variables.

Definition

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The cumulative distribution function of a real-valuedrandom variable $X {\displaystyle X}$ is the function given by^[2]^: 77

$F_{X}(x)=\operatorname {P} (X\leq x)$ (Eq.1)

where the right-hand side represents theprobability that the random variable $X {\displaystyle X}$ takes on a value less than or equal to $x {\displaystyle x}$ .

The probability that $X {\displaystyle X}$ lies in the semi-closedinterval $(a,b]$ , where $a<b$ , is therefore^[2]^: 84

$\operatorname {P} (a<X\leq b)=F_{X}(b)-F_{X}(a)$ (Eq.2)

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 the distinction is important for discrete distributions. The proper use of tables of thebinomial andPoisson distributions depends upon this convention. Moreover, important formulas likePaul Lévy's inversion formula for thecharacteristic function also rely on the "less than or equal" formulation.

If treating several random variables $X,Y,\ldots$ 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 $F {\displaystyle F}$ for a cumulative distribution function, in contrast to the lower-case $f {\displaystyle f}$ used forprobability density functions andprobability mass functions. This applies when discussing general distributions: some specific distributions have their own conventional notation, for example thenormal distribution uses $\Phi$ and $\phi$ instead of $F {\displaystyle F}$ and $f {\displaystyle f}$ , respectively.

The probability density function of a continuous random variable can be determined from the cumulative distribution function by differentiating^[3] using theFundamental Theorem of Calculus; i.e. given $F(x)$ , $f(x)={\frac {dF(x)}{dx}}$ as long as the derivative exists.

The CDF of acontinuous random variable $X {\displaystyle X}$ can be expressed as the integral of its probability density function $f_{X}$ as follows:^[2]^: 86 $F_{X}(x)=\int _{-\infty }^{x}f_{X}(t)\,dt.$

In the case of a random variable $X {\displaystyle X}$ which has distribution having a discrete component at a value $b {\displaystyle b}$ , $\operatorname {P} (X=b)=F_{X}(b)-\lim _{x\to b^{-}}F_{X}(x).$

If $F_{X}$ is continuous at $b {\displaystyle b}$ , this equals zero and there is no discrete component at $b {\displaystyle b}$ .

Properties

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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.

Example of a cumulative distribution function with a countably infinite set of discontinuities.

Every cumulative distribution function $F_{X}$ isnon-decreasing^[2]^: 78 andright-continuous,^[2]^: 79 which makes it acàdlàg function. Furthermore, $\lim _{x\to -\infty }F_{X}(x)=0,\quad \lim _{x\to +\infty }F_{X}(x)=1.$

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

If $X {\displaystyle X}$ is a purelydiscrete random variable, then it attains values $x_{1},x_{2},\ldots$ with probability $p_{i}=p(x_{i})$ , and the CDF of $X {\displaystyle X}$ will bediscontinuous at the points $x_{i}$ : $F_{X}(x)=\operatorname {P} (X\leq x)=\sum _{x_{i}\leq x}\operatorname {P} (X=x_{i})=\sum _{x_{i}\leq x}p(x_{i}).$

If the CDF $F_{X}$ of a real valued random variable $X {\displaystyle X}$ iscontinuous, then $X {\displaystyle X}$ is acontinuous random variable; if furthermore $F_{X}$ isabsolutely continuous, then there exists aLebesgue-integrable function $f_{X}(x)$ such that $F_{X}(b)-F_{X}(a)=\operatorname {P} (a<X\leq b)=\int _{a}^{b}f_{X}(x)\,dx$ for all real numbers $a {\displaystyle a}$ and $b {\displaystyle b}$ . The function $f_{X}$ is equal to thederivative of $F_{X}$ almost everywhere, and it is called theprobability density function of the distribution of $X {\displaystyle X}$ .

If $X {\displaystyle X}$ has finiteL1-norm, that is, the expectation of $|X|$ is finite, then the expectation is given by theRiemann–Stieltjes integral $\mathbb {E} [X]=\int _{-\infty }^{\infty }t\,dF_{X}(t)$

CDF plot with two red rectangles, illustrating two inequalities

and for any $x\geq 0$ , $x(1-F_{X}(x))\leq \int _{x}^{\infty }t\,dF_{X}(t)$ as well as $xF_{X}(-x)\leq \int _{-\infty }^{-x}(-t)\,dF_{X}(t)$ as shown in the diagram (consider the areas of the two red rectangles and their extensions to the right or left up to the graph of $F_{X}$ ).^{[clarification needed]} In particular, we have $\lim _{x\to -\infty }xF_{X}(x)=0,\quad \lim _{x\to +\infty }x(1-F_{X}(x))=0.$ In addition, the (finite) expected value of the real-valued random variable $X {\displaystyle X}$ can be defined on the graph of its cumulative distribution function as illustrated by thedrawing in thedefinition of expected value for arbitrary real-valued random variables.

Examples

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As an example, suppose $X {\displaystyle X}$ isuniformly distributed on the unit interval $[0,1]$ .

Then the CDF of $X {\displaystyle X}$ is given by $F_{X}(x)={\begin{cases}0&:\ x<0\\x&:\ 0\leq x\leq 1\\1&:\ x>1\end{cases}}$

Suppose instead that $X {\displaystyle X}$ takes only the discrete values 0 and 1, with equal probability.

Then the CDF of $X {\displaystyle X}$ is given by $F_{X}(x)={\begin{cases}0&:\ x<0\\1/2&:\ 0\leq x<1\\1&:\ x\geq 1\end{cases}}$

Suppose $X {\displaystyle X}$ isexponential distributed. Then the CDF of $X {\displaystyle X}$ is given by $F_{X}(x;\lambda )={\begin{cases}1-e^{-\lambda x}&x\geq 0,\\0&x<0.\end{cases}}$

Hereλ > 0 is the parameter of the distribution, often called the rate parameter.

Suppose $X {\displaystyle X}$ isnormal distributed. Then the CDF of $X {\displaystyle X}$ is given by $F(t;\mu ,\sigma )={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{t}\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right)\,dx.$

Here the parameter $\mu$ is themean or expectation of the distribution; and $\sigma$ is its standard deviation.

A table of the CDF of the standard normal distribution is often used in statistical applications, where it is named thestandard normal table, theunit normal table, or theZ table.

Suppose $X {\displaystyle X}$ isbinomial distributed. Then the CDF of $X {\displaystyle X}$ is given by $F(k;n,p)=\Pr(X\leq k)=\sum _{i=0}^{\lfloor k\rfloor }{n \choose i}p^{i}(1-p)^{n-i}$

Here $p {\displaystyle p}$ is the probability of success and the function denotes the discrete probability distribution of the number of successes in a sequence of $n {\displaystyle n}$ independent experiments, and $\lfloor k\rfloor$ is the "floor" under $k {\displaystyle k}$ , i.e. the greatest integer less than or equal to $k {\displaystyle k}$ .

Derived functions

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Complementary cumulative distribution function (tail distribution)

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Sometimes, it is useful to study the opposite question and ask how often the random variable isabove a particular level. This is called thecomplementary cumulative distribution function (ccdf) or simply thetail distribution orexceedance, and is defined as ${\bar {F}}_{X}(x)=\operatorname {P} (X>x)=1-F_{X}(x).$

This has applications instatistical hypothesis testing, for example, because the one-sidedp-value is the probability of observing a test statisticat least as extreme as the one observed. Thus, provided that thetest statistic,T, has a continuous distribution, the one-sidedp-value is simply given by the ccdf: for an observed value $t {\displaystyle t}$ of the test statistic $p=\operatorname {P} (T\geq t)=\operatorname {P} (T>t)=1-F_{T}(t).$

Insurvival analysis, ${\bar {F}}_{X}(x)$ is called thesurvival function and denoted $S(x)$ , while the termreliability function is common inengineering.

Properties

For a non-negative continuous random variable having an expectation,Markov's inequality states that^[4] ${\bar {F}}_{X}(x)\leq {\frac {\operatorname {E} (X)}{x}}.$
As $x\to \infty ,{\bar {F}}_{X}(x)\to 0$ , and in fact ${\bar {F}}_{X}(x)=o(1/x)$ provided that $\operatorname {E} (X)$ is finite.
Proof:^{[citation needed]}
Assuming $X {\displaystyle X}$ has a density function $f_{X}$ , for any $c>0$ $\operatorname {E} (X)=\int _{0}^{\infty }xf_{X}(x)\,dx\geq \int _{0}^{c}xf_{X}(x)\,dx+c\int _{c}^{\infty }f_{X}(x)\,dx$ Then, on recognizing ${\bar {F}}_{X}(c)=\int _{c}^{\infty }f_{X}(x)\,dx$ and rearranging terms, $0\leq c{\bar {F}}_{X}(c)\leq \operatorname {E} (X)-\int _{0}^{c}xf_{X}(x)\,dx\to 0{\text{ as }}c\to \infty$ as claimed.
For a random variable having an expectation, $\operatorname {E} (X)=\int _{0}^{\infty }{\bar {F}}_{X}(x)\,dx-\int _{-\infty }^{0}F_{X}(x)\,dx$ and for a non-negative random variable the second term is 0.
If the random variable can only take non-negative integer values, this is equivalent to $\operatorname {E} (X)=\sum _{n=0}^{\infty }{\bar {F}}_{X}(n).$

Folded cumulative distribution

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While the plot of a cumulative distribution $F {\displaystyle F}$ often has an S-like shape, an alternative illustration is thefolded cumulative distribution ormountain plot, which folds the top half of the graph over,^[5]^[6] that is

F_{\text{fold}}(x)=F(x)1_{\{F(x)\leq 0.5\}}+(1-F(x))1_{\{F(x)>0.5\}}

where $1_{\{A\}}$ denotes theindicator function and the second summand is thesurvivor function, thus using two scales, one for the upslope and another for the downslope. This form of illustration emphasises themedian,dispersion (specifically, themean absolute deviation from the median^[7]) andskewness of the distribution or of the empirical results.

Inverse distribution function (quantile function)

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Main article:Quantile function

If the CDFF is strictly increasing and continuous then $F^{-1}(p),p\in [0,1],$ is the unique real number $x {\displaystyle x}$ such that $F(x)=p$ . This defines theinverse distribution function orquantile function.

Some distributions do not have a unique inverse (for example if $f_{X}(x)=0$ for all $a<x<b$ , causing $F_{X}$ to be constant). In this case, one may use thegeneralized inverse distribution function, which is defined as

F^{-1}(p)=\inf\{x\in \mathbb {R} :F(x)\geq p\},\quad \forall p\in [0,1].

Example 1: The median is $F^{-1}(0.5)$ .
Example 2: Put $\tau =F^{-1}(0.95)$ . Then we call $\tau$ the 95th percentile.

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

$F^{-1}$ is nondecreasing^[8]
$F^{-1}(F(x))\leq x$
$F(F^{-1}(p))\geq p$
$F^{-1}(p)\leq x$ if and only if $p\leq F(x)$
If $Y {\displaystyle Y}$ has a $U[0,1]$ distribution then $F^{-1}(Y)$ is distributed as $F {\displaystyle F}$ . This is used inrandom number generation using theinverse transform sampling-method.
If $\{X_{\alpha }\}$ is a collection of independent $F {\displaystyle F}$ -distributed random variables defined on the samesample space, then there exist random variables $Y_{\alpha }$ such that $Y_{\alpha }$ is distributed as $U[0,1]$ and $F^{-1}(Y_{\alpha })=X_{\alpha }$ with probability 1 for all $\alpha$ .^{[citation needed]}

The inverse of the cdf can be used to translate results obtained for the uniform distribution to other distributions.

Empirical distribution function

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Theempirical distribution function is an estimate of the cumulative distribution function that generated the points in the sample. It converges with probability 1 to that underlying distribution. A number of results exist to quantify therate of convergence of the empirical distribution function to the underlying cumulative distribution function.^[9]

Multivariate case

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Definition for two random variables

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When dealing simultaneously with more than one random variable thejoint cumulative distribution function can also be defined. For example, for a pair of random variables $X, Y {\displaystyle X,Y}$ , the joint CDF $F_{XY}$ is given by^[2]^: 89

$F_{X,Y}(x,y)=\operatorname {P} (X\leq x,Y\leq y)$ (Eq.3)

where the right-hand side represents theprobability that the random variable $X {\displaystyle X}$ takes on a value less than or equal to $x {\displaystyle x}$ and that $Y {\displaystyle Y}$ takes on a value less than or equal to $y {\displaystyle y}$ .

Example of joint cumulative distribution function:

For two continuous variablesX andY: $\Pr(a<X<b{\text{ and }}c<Y<d)=\int _{a}^{b}\int _{c}^{d}f(x,y)\,dy\,dx;$

For two discrete random variables, it is beneficial to generate a table of probabilities and address the cumulative probability for each potential range ofX andY, and here is the example:^[10]

given the joint probability mass function in tabular form, determine the joint cumulative distribution function.

	Y = 2	Y = 4	Y = 6	Y = 8
X = 1	0	0.1	0	0.1
X = 3	0	0	0.2	0
X = 5	0.3	0	0	0.15
X = 7	0	0	0.15	0

Solution: using the given table of probabilities for each potential range ofX andY, the joint cumulative distribution function may be constructed in tabular form:

	Y < 2	Y ≤ 2	Y ≤ 4	Y ≤ 6	Y ≤ 8
X < 1	0	0	0	0	0
X ≤ 1	0	0	0.1	0.1	0.2
X ≤ 3	0	0	0.1	0.3	0.4
X ≤ 5	0	0.3	0.4	0.6	0.85
X ≤ 7	0	0.3	0.4	0.75	1

Definition for more than two random variables

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For $N {\displaystyle N}$ random variables $X_{1},\ldots ,X_{N}$ , the joint CDF $F_{X_{1},\ldots ,X_{N}}$ is given by

$F_{X_{1},\ldots ,X_{N}}(x_{1},\ldots ,x_{N})=\operatorname {P} (X_{1}\leq x_{1},\ldots ,X_{N}\leq x_{N})$ (Eq.4)

Interpreting the $N {\displaystyle N}$ random variables as arandom vector $\mathbf {X} =(X_{1},\ldots ,X_{N})^{T}$ yields a shorter notation: $F_{\mathbf {X} }(\mathbf {x} )=\operatorname {P} (X_{1}\leq x_{1},\ldots ,X_{N}\leq x_{N})$

Properties

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Every multivariate CDF is:

Monotonically non-decreasing for each of its variables,
Right-continuous in each of its variables,
$0\leq F_{X_{1}\ldots X_{n}}(x_{1},\ldots ,x_{n})\leq 1,$
$\lim _{x_{1},\ldots ,x_{n}\to +\infty }F_{X_{1}\ldots X_{n}}(x_{1},\ldots ,x_{n})=1$ and $\lim _{x_{i}\to -\infty }F_{X_{1}\ldots X_{n}}(x_{1},\ldots ,x_{n})=0,$ for alli.

Not every function satisfying the above four properties is a multivariate CDF, unlike in the single dimension case. For example, let $F(x,y)=0$ for $x<0$ or $x+y<1$ or $y<0$ and let $F(x,y)=1$ otherwise. It is easy to see that the above conditions are met, and yet $F {\displaystyle F}$ is not a CDF since if it was, then ${\textstyle \operatorname {P} \left({\frac {1}{3}}<X\leq 1,{\frac {1}{3}}<Y\leq 1\right)=-1}$ as explained below.

The probability that a point belongs to ahyperrectangle is analogous to the 1-dimensional case:^[11] $F_{X_{1},X_{2}}(a,c)+F_{X_{1},X_{2}}(b,d)-F_{X_{1},X_{2}}(a,d)-F_{X_{1},X_{2}}(b,c)=\operatorname {P} (a<X_{1}\leq b,c<X_{2}\leq d)=\int \cdots$

Complex case

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Complex random variable

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The generalization of the cumulative distribution function from real tocomplex random variables is not obvious because expressions of the form $P(Z\leq 1+2i)$ make no sense. However expressions of the form $P(\Re {(Z)}\leq 1,\Im {(Z)}\leq 3)$ make sense. Therefore, we define the cumulative distribution of a complex random variables via thejoint distribution of their real and imaginary parts: $F_{Z}(z)=F_{\Re {(Z)},\Im {(Z)}}(\Re {(z)},\Im {(z)})=P(\Re {(Z)}\leq \Re {(z)},\Im {(Z)}\leq \Im {(z)}).$

Complex random vector

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Generalization ofEq.4 yields ${\begin{aligned}F_{\mathbf {Z} }(\mathbf {z} )&=F_{\Re {(Z_{1})},\Im {(Z_{1})},\ldots ,\Re {(Z_{n})},\Im {(Z_{n})}}(\Re {(z_{1})},\Im {(z_{1})},\ldots ,\Re {(z_{n})},\Im {(z_{n})})\\[1ex]&=\operatorname {P} (\Re {(Z_{1})}\leq \Re {(z_{1})},\Im {(Z_{1})}\leq \Im {(z_{1})},\ldots ,\Re {(Z_{n})}\leq \Re {(z_{n})},\Im {(Z_{n})}\leq \Im {(z_{n})})\end{aligned}}$ as definition for the CDS of a complex random vector $\mathbf {Z} =(Z_{1},\ldots ,Z_{N})^{T}$ .

Use in statistical analysis

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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. Theempirical 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 variousstatistical hypothesis tests. 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

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TheKolmogorov–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 relatedKuiper'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.

References

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^Deisenroth, Marc Peter; Faisal, A. Aldo; Ong, Cheng Soon (2020).Mathematics for Machine Learning. Cambridge University Press. p. 181.ISBN 9781108455145.
^^a ^b ^c ^d ^e ^fPark, Kun Il (2018).Fundamentals of Probability and Stochastic Processes with Applications to Communications. Springer.ISBN 978-3-319-68074-3.
^Montgomery, Douglas C.; Runger, George C. (2003).Applied Statistics and Probability for Engineers(PDF). John Wiley & Sons, Inc. p. 104.ISBN 0-471-20454-4.Archived(PDF) from the original on 2012-07-30.
^Zwillinger, Daniel; Kokoska, Stephen (2010).CRC Standard Probability and Statistics Tables and Formulae. CRC Press. p. 49.ISBN 978-1-58488-059-2.
^Gentle, J.E. (2009).Computational Statistics.Springer.ISBN 978-0-387-98145-1. Retrieved2010-08-06.^{[page needed]}
^Monti, K. L. (1995). "Folded Empirical Distribution Function Curves (Mountain Plots)".The American Statistician.49 (4):342–345.doi:10.2307/2684570.JSTOR 2684570.
^Xue, J. H.; Titterington, D. M. (2011)."The p-folded cumulative distribution function and the mean absolute deviation from the p-quantile"(PDF).Statistics & Probability Letters.81 (8):1179–1182.doi:10.1016/j.spl.2011.03.014.
^Chan, Stanley H. (2021).Introduction to Probability for Data Science. Michigan Publishing. p. 18.ISBN 978-1-60785-746-4.
^Hesse, C. (1990). "Rates of convergence for the empirical distribution function and the empirical characteristic function of a broad class of linear processes".Journal of Multivariate Analysis.35 (2):186–202.doi:10.1016/0047-259X(90)90024-C.
^"Joint Cumulative Distribution Function (CDF)".math.info. Retrieved2019-12-11.
^"Archived copy"(PDF).www.math.wustl.edu. Archived fromthe original(PDF) on 22 February 2016. Retrieved13 January 2022.{{cite web}}: CS1 maint: archived copy as title (link)

External links

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Media related toCumulative distribution functions at Wikimedia Commons

v t e Theory ofprobability distributions
probability mass function (pmf) probability density function (pdf) cumulative distribution function (cdf) quantile function
raw moment central moment mean variance standard deviation skewness kurtosis L-moment
moment-generating function (mgf) characteristic function probability-generating function (pgf) cumulant combinant