Difference between revisions of "Probability density function"
(New page: In probability theory, a ''probability density function'' (say) ''f'' is a real valued and continuous function whose value is the probability density of the variable that it is a funct...) |
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:<math>P(a \leq x \leq b) = \int_a^b f(x) \, dx </math> | :<math>P(a \leq x \leq b) = \int_a^b f(x) \, dx </math> | ||
| − | This density function is intended to express mathematically the total apportionment of the values of the variable it represents over its entire [[domain]]. In order to qualify, such a function must meet the following criteria: | + | This density function is intended to express mathematically the total apportionment of the values of the variable it represents over its entire [[domain]]. It is necessary to express probability this way for a variable or parameter which may take on a continuum of values, so that the total probability covering the entire domain of support may converge to a finite value. The counterpart for a discretely distributed variable is the [[probability mass function]]. |
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| + | In order to qualify, such a function must meet the following criteria: | ||
(1) <math> \int_{-\infty}^\infty \,f(x)\,dx = 1. </math> i.e., finitely convergent (to unity by convention). | (1) <math> \int_{-\infty}^\infty \,f(x)\,dx = 1. </math> i.e., finitely convergent (to unity by convention). | ||
(2) <math> \int_{-\infty}^a \,f(x)\,dx \leq \int_{-\infty}^b \,f(x)\,dx </math> for a<b, i.e., is non-decreasing | (2) <math> \int_{-\infty}^a \,f(x)\,dx \leq \int_{-\infty}^b \,f(x)\,dx </math> for a<b, i.e., is non-decreasing | ||
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| + | Such a function leads to the definition of an associated [[cumulative distribution function]]. | ||
[[Category:mathematics]] | [[Category:mathematics]] | ||
Revision as of 13:46, December 13, 2007
In probability theory, a probability density function (say) f is a real valued and continuous function whose value is the probability density of the variable that it is a function of. Since it is a density, the actual probability P that the variable will be in the interval [a,b] is
- <math>P(a \leq x \leq b) = \int_a^b f(x) \, dx </math>
This density function is intended to express mathematically the total apportionment of the values of the variable it represents over its entire domain. It is necessary to express probability this way for a variable or parameter which may take on a continuum of values, so that the total probability covering the entire domain of support may converge to a finite value. The counterpart for a discretely distributed variable is the probability mass function.
In order to qualify, such a function must meet the following criteria:
(1) <math> \int_{-\infty}^\infty \,f(x)\,dx = 1. </math> i.e., finitely convergent (to unity by convention).
(2) <math> \int_{-\infty}^a \,f(x)\,dx \leq \int_{-\infty}^b \,f(x)\,dx </math> for a<b, i.e., is non-decreasing
Such a function leads to the definition of an associated cumulative distribution function.