Definition:Cumulative Distribution Function

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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

  • Results about cumulative distribution functions can be found here.