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 distribution function 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

• Results about cumulative distribution functions can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): cumulative distribution function
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): distribution function
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cumulative frequency function
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): distribution function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cumulative frequency function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): distribution function
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): cumulative distribution function
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): distribution function