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Cumulative distribution function (view source)
Revision as of 07:12, July 13, 2016
, 07:12, July 13, 2016→top: clean up & uniformity
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.,
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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 of the real line up and including the value of that variable; i.e.,
:<math> F(x) = \int_{-\infty}^x \,f(\lambda)\,d\lambda</math>
:<math> F(x) = \int_{-\infty}^x \,f(\lambda)\,d\lambda</math>
Considering this definition in light of the [[Fundamental Theorem of Calculus]] yields:
:<math> f(x) = \frac{dF(x)}{dx}</math>
(2) <math> F(\infty) = 1 </math>, i.e., finitely convergent (to unity by convention).
(2) <math> \lim_{x \to \infty}F(x) = 1 </math>, i.e., finitely convergent (to unity by convention).
If the [[domain]] of the variable is [[finite]], then the argument in equation (2) above should be the upper bounds.
If the [[domain]] of the variable is [[finite]], then the upper limit in equation (2) above should be the upper bound of the variables [[domain of support]].
[[Category:mathematics]]
[[Category:Probability and Statistics]]