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(Add more examples both in one and two variables case; add sentences to explain how to get PDF from CDF; add derived function like Ztable and empirical function.) 

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 lowercase <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
[[Probability density functionProbability Density Function]] from the Cumulative Distribution Function<ref>{{Cite booktitle=Applied Statistics and Probability for Engineerslast=first=publisher=year=isbn=1119456266location=pages=70}}</ref>
Given F(x),
''f (x) ='' <math>{dF(x) \over dx}</math>, as long as the derivative exists.
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" />{{rpp. 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>
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 [[
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>
Suppose <math>X</math> is [[Exponential distributionexponential distributed]]. Then the CDF of <math>X</math> is given by
: <math>F_X(x;\lambda) = \begin{cases}
1e^{\lambda x} & x \ge 0, \\
0 & x < 0.
\end{cases}</math>
Here λ > 0 is the parameter of the distribution, often called the rate parameter.
Suppose <math>X</math> is [[Normal distributionnormal distributed]]. Then the CDF of <math>X</math> is given by
: <math>
F(x;\mu,\sigma)
=
\frac{1}{\sigma\sqrt{2\pi}}
\int_{\infty}^x
\exp
\left( \frac{(t  \mu)^2}{2\sigma^2}
\ \right)\, dt.
</math>
Here the parameter <math>\mu</math> is the mean or expectation of the distribution; and <math>\sigma</math> is its standard deviation.
Suppose <math>X</math> is [[Binomial distributionbinomial distributed]]. Then the CDF of <math>X</math> is given by
: <math>F(k;n,p)=\Pr(X\leq k)=\sum _{i=0}^{\lfloor k\rfloor }{n \choose i}p^{i}(1p)^{ni}</math>
Here parameters <math>n</math> and <math>p</math> is the discrete probability distribution of the number of successes in a sequence of n independent experiments, and <math>\lfloor k\rfloor\,</math> is the "floor" under <math>k</math>, i.e. the [[greatest integer]] less than or equal to <math>k</math>.
<br />
==Derived functions==
===Complementary cumulative distribution function (tail distribution)===<! This section is linked from [[Power law]], [[Stretched exponential function]] and [[Weibull distribution]] >
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]].
Ztable:
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 weburl=https://www.ztable.net/title=Z Tablelast=first=date=website=Z Tablelanguage=enUSurlstatus=livearchiveurl=archivedate=accessdate=20191211}}</ref>, is the value of cumulative distribution function of the normal distribution. It is very useful to use Ztable 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.
<br />
;Properties
# 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.
<br />
=== '''Empirical distribution function''' ===
The [[empirical 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 the rate of convergence of the empirical distribution function to the underlying cumulative distribution function.
==Multivariate case==
equal to <math>y</math>.
Example of joint cumulative distribution function:
For two continuous variables X and Y: P((a<X<b)and(c<Y<d))=<math>\int\limits_{a}^{b} \int\limits_{c}^{d} f(x,y)dy dx</math>;
For two discrete random variables, it is beneficial to generate a table of probabilities and address the cumulative probability for each potential range of X and Y, and here is the example<ref>{{Cite weburl=https://math.info/Probability/Joint_CDF/title=Joint Cumulative Density Function (CDF)website=math.infoaccessdate=20191211}}</ref>:
given the joint probability density function in tabular form, determine the joint cumulative distribution function.
{ class="wikitable"

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 of X and Y, the joint cumulative distribution function may be constructed in tabular form:
{ class="wikitable"

Y<2
2≤Y<4
4≤Y<6
6≤Y<8
Y≤8

X<1
0
0
0
0
0

1≤X<3
0
0
0.1
0.1
0.2

3≤X<5
0
0
0.1
0.3
0.4

5≤X<7
0
0.3
0.4
0.6
0.85

X≤7
0
0.3
0.4
0.75
1
}
<br />
===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

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