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., | + | 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., |
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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 equation (2) above should be the upper bound. |
[[Category:mathematics]] | [[Category:mathematics]] | ||
Revision as of 14:15, December 31, 2007
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.,
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x) = \int_{-\infty}^x \,f(\lambda)\,d\lambda}
Due to the properties of the probability density function f(x), the cumulative distribution function F(x) will have the following properties:
(1) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x) \geq 0 }
inside its domain of support.
(2) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to \infty}F(x) = 1 }
, i.e., finitely convergent (to unity by convention).
(3) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(a) \leq F(b) }
for a<b, i.e., is non-decreasing
If the domain of the variable is finite, then the upper limit equation (2) above should be the upper bound.