Distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable



X


{\displaystyle X}
, or just distribution function of



X


{\displaystyle X}
, evaluated at



x


{\displaystyle x}
, is the probability that



X


{\displaystyle X}
will take a value less than or equal to



x


{\displaystyle x}
.Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by an upwards continuous monotonic increasing cumulative distribution function



F
:

R

→
[
0
,
1
]


{\displaystyle F:\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 continuous distribution, it gives the area under the probability density function from minus infinity to



x


{\displaystyle x}
. Cumulative distribution functions are also used to specify the distribution of multivariate random variables.

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