Cumulative distribution function

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a> and <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the cumulative distribution function (CDF) of a real-valued <a href="/facts/Random_variable/TwTBXnLT">random variable</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
, or just distribution function of 
 
 
 
 X
 
 
 {\displaystyle X}
 
, evaluated at 
 
 
 
 x
 
 
 {\displaystyle x}
 
, is the <a href="/facts/Probability/fgYvMhwb">probability</a> that 
 
 
 
 X
 
 
 {\displaystyle X}
 
 will take a value less than or equal to 
 
 
 
 x
 
 
 {\displaystyle x}
 
.
Every <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> <a href="/facts/Support_(measure_theory)/mBM9wEAF">supported</a> on the real numbers, discrete or "mixed" as well as <a href="/facts/Continuous_variable/TWl5feEw">continuous</a>, is uniquely identified by a <a href="/facts/Right-continuous/sKbl02pB">right-continuous</a> <a href="/facts/Monotonic_function/MjQK0YL2">monotone increasing</a> function (a <a href="/facts/C%C3%A0dl%C3%A0g/R9Jz9QLh">càdlàg</a> function) 
 
 
 
 F
 :
 
 R
 
 →
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle F\colon \mathbb {R} \rightarrow [0,1]}
 
 satisfying 
 
 
 
 
 lim
 
 x
 →
 −
 ∞
 
 
 F
 (
 x
 )
 =
 0
 
 
 {\displaystyle \lim _{x\rightarrow -\infty }F(x)=0}
 
 and 
 
 
 
 
 lim
 
 x
 →
 ∞
 
 
 F
 (
 x
 )
 =
 1
 
 
 {\displaystyle \lim _{x\rightarrow \infty }F(x)=1}
 
.
In the case of a scalar <a href="/facts/Continuous_distribution/EpsKKVRu">continuous distribution</a>, it gives the area under the <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> from negative infinity to 
 
 
 
 x
 
 
 {\displaystyle x}
 
. Cumulative distribution functions are also used to specify the distribution of <a href="/facts/Multivariate_random_variable/qMfooyVf">multivariate random variables</a>.

Cumulative distribution function open-in-new

Cumulative distribution function