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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 ${\displaystyle x}$.^[1]

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) ${\displaystyle F\colon \mathbb {R} \rightarrow }$ 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.

Definition

The cumulative distribution function of a real-valued random variable ${\displaystyle X}$ is the function given by^{[2]: p. 77}

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

(Eq.1)

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^{[2]: p. 84}

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

(Eq.2)

In the definition above, the "less than or equal to" sign, "≤", is a convention, not a universally used one (e.g. Hungarian literature uses "<"), but the distinction is important for discrete distributions. The proper use of tables of the binomial and Poisson distributions depends upon this convention. Moreover, important formulas like Paul Lévy's inversion formula for the characteristic function also rely on the "less than or equal" formulation.

If treating several random variables ${\displaystyle X,Y,\ldots }$ 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 ${\displaystyle F}$ for a cumulative distribution function, in contrast to the lower-case ${\displaystyle f}$ used for probability density functions and probability mass functions. This applies when discussing general distributions: some specific distributions have their own conventional notation, for example the normal distribution uses ${\displaystyle \Phi }$ and ${\displaystyle \phi }$ instead of ${\displaystyle F}$ and ${\displaystyle f}$, respectively.

The probability density function of a continuous random variable can be determined from the cumulative distribution function by differentiating^[3] using the Fundamental Theorem of Calculus; i.e. given ${\displaystyle F(x)}$,

$${\displaystyle f(x)={\frac {dF(x)}{dx}}}$$

The CDF of a continuous random variable ${\displaystyle X}$ can be expressed as the integral of its probability density function ${\displaystyle f_{X}}$ as follows:^{[2]: p. 86}

$${\displaystyle F_{X}(x)=\int _{-\infty }^{x}f_{X}(t)\,dt.}$$

In the case of a random variable ${\displaystyle X}$ which has distribution having a discrete component at a value ${\displaystyle b}$,

$${\displaystyle \operatorname {P} (X=b)=F_{X}(b)-\lim _{x\to b^{-}}F_{X}(x).}$$

If ${\displaystyle F_{X}}$ is continuous at ${\displaystyle b}$, this equals zero and there is no discrete component at ${\displaystyle b}$.

Properties

Every cumulative distribution function ${\displaystyle F_{X}}$ is non-decreasing^{[2]: p. 78} and right-continuous,^{[2]: p. 79} which makes it a càdlàg function. Furthermore,