Difference between revisions of "Cumulative distribution function"
(New 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...) |
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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 real line up and including the value of that variable; i.e., | + | {|align="right" border="1" |
| + | |+A Cumulative Distribution Function | ||
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| + | |[[Image:Law-uniform.png|px=133]] | ||
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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> | ||
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| + | Considering this definition in light of the [[Fundamental Theorem of Calculus]] yields: | ||
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| + | :<math> f(x) = \frac{dF(x)}{dx}</math> | ||
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| − | (2) <math> F( | + | (2) <math> \lim_{x \to \infty}F(x) = 1 </math>, i.e., finitely convergent (to unity by convention). |
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| − | If the [[domain]] of the variable is [[finite]], then the | + | 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]]. |
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| − | [[Category: | + | [[Category:Probability and Statistics]] |
Latest revision as of 07:12, July 13, 2016
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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>
Considering this definition in light of the Fundamental Theorem of Calculus yields:
- <math> f(x) = \frac{dF(x)}{dx}</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> \lim_{x \to \infty}F(x) = 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 upper limit in equation (2) above should be the upper bound of the variables domain of support.