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:
- 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
- Survival Function, a closely related concept
- Distribution Function of Finite Signed Borel Measure, of which this is an instantiation
- 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
- 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