real numbers cumulative distribution function random variable subset sample space real number probability
Modern definition:
If the sample space of a random variable X is the set of real numbers (




R



{\displaystyle \mathbb {R} }

) or a subset thereof, then a function called the cumulative distribution function (CDF) 



F



{\displaystyle F\,}

 exists, defined by 



F
(
x
)
=
P
(
X
≤
x
)



{\displaystyle F(x)=P(X\leq x)\,}

. That is, F(x) returns the probability that X will be less than or equal to x.
