In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable ${\displaystyle X}$, or just distribution function of ${\displaystyle X}$, evaluated at ${\displaystyle x}$, is the probability that ${\displaystyle X}$ will take a value less than or equal to ${\displaystyle x}$.

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

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

The cumulative distribution function of a real-valued random variable ${\displaystyle X}$ is the function given by

${\displaystyle F_{X}(x)=\operatorname {P} (X\leq x)}$   ()

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

The probability that ${\displaystyle X}$ lies in the semi-closed interval ${\displaystyle (a,b]}$, where ${\displaystyle a<b}$, is therefore

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