Definition:Cumulative Distribution Function

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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 of $X$ is denoted $F_X$, and defined as:

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


Examples

Arbitrary Example

Consider a population consisting of children the state of whose teeth is being monitored.

The following table consists of a count of the number of teeth with dental caries in a group of $50$ schoolchildren:

$\begin {array} {|l|l|}

\hline \text {Number of Teeth} & \text {Number of Children} \\ \hline 0 & 27 \\ 1 & 12 \\ 2 & 6 \\ 3 & 4 \\ 6 & 1 \\ \hline \end {array}$

The values of the cumulative distribution function:

\(\ds \map F 0\) \(=\) \(\ds \dfrac {27} {50}\)
\(\ds \map F 1\) \(=\) \(\ds \dfrac {39} {50}\)
\(\ds \map F 2\) \(=\) \(\ds \dfrac {45} {50}\)
\(\ds \map F 3\) \(=\) \(\ds \dfrac {49} {50}\)
\(\ds \map F 4\) \(=\) \(\ds \dfrac {49} {50}\)
\(\ds \map F 5\) \(=\) \(\ds \dfrac {49} {50}\)
\(\ds \map F 6\) \(=\) \(\ds 1\)


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.


The abbreviation c.d.f. or cdf are often used.


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


Also see

  • Results about cumulative distribution functions can be found here.


Sources

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