# Definition:Cumulative Distribution Function

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## Definition

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

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

The **cumulative distribution function** (or **c.d.f.**) of $X$ is denoted $\map F X$, and defined as:

- $\forall x \in \R: \map {\map F X} x := \map \Pr {X \le x}$

## Also known as

Some sources refer to this as a **distribution function**, but it can then become confused with the concept of a distribution function in physics.

Others use the term **probability distribution**.

Some sources use the notation $\map \Phi X$ for $\map F X$.

## Also see

- Survival Function, a closely related concept

- Results about
**cumulative distribution functions**can be found here.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**cumulative distribution function**