Definition:Cumulative Distribution Function

Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.

The cumulative distribution function (or c.d.f.) of $X$ is denoted $F_X$, and defined as:
 * $\forall x \in \R: \map {F_X} x := \map \Pr {X \le x}$

Also known as
Other terms used for cumulative distribution function:


 * Probability distribution


 * Cumulative frequency function


 * Distribution function, but this can then become confused with the concept of a in physics.

Some sources use the notation $\Phi_X$, $\map \Phi X$ or $\map F X$ for $F_X$.

Also see

 * Survival Function, a closely related concept


 * Distribution Function of Finite Signed Borel Measure, of which this is an instantiation


 * Properties of Cumulative Distribution Function