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Cumulative distribution function (view source)
Revision as of 22:47, December 30, 2007
, 22:47, December 30, 2007New page: In probability theory, a ''cumulative distribution function'' ''F(x)'' of a probability density function say ''f(x)'' is a real valued and continuous function whose value is the p...
In [[probability theory]], a ''cumulative distribution function'' ''F(x)'' of a [[probability density function]] say ''f(x)'' is a real valued and continuous function whose value is the proportion of probability values of a variable which occur on the part real line up and including the value of that variable; i.e.,
:<math> F(x) = \int_{-\infty}^x \,f(\lambda)\,d\lambda</math>
Due to the properties of the [[probability density function]] ''f(x)'', the cumulative distribution function ''F(x)'' will have the following properties:
(1) <math> F(x) \geq 0 </math> <math> \forall x </math> inside its domain of support.
(2) <math> F(\infty) = 1 </math>, i.e., finitely convergent (to unity by convention).
(3) <math> F(a) \leq F(b) </math> for a<b, i.e., is non-decreasing
If the [[domain]] of the variable is [[finite]], then the argument in equation (2) above should be the upper bounds.
[[Category:mathematics]]
:<math> F(x) = \int_{-\infty}^x \,f(\lambda)\,d\lambda</math>
Due to the properties of the [[probability density function]] ''f(x)'', the cumulative distribution function ''F(x)'' will have the following properties:
(1) <math> F(x) \geq 0 </math> <math> \forall x </math> inside its domain of support.
(2) <math> F(\infty) = 1 </math>, i.e., finitely convergent (to unity by convention).
(3) <math> F(a) \leq F(b) </math> for a<b, i.e., is non-decreasing
If the [[domain]] of the variable is [[finite]], then the argument in equation (2) above should be the upper bounds.
[[Category:mathematics]]