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mLine 4:
Line 4: − This density function is intended to express mathematically the total apportionment of the values of the variable it represents over the variables 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]]. − + + + + + + +
Probability density function (view source)
Revision as of 14:59, December 13, 2007
, 14:59, December 13, 2007no edit summary
:<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>
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 the variables entire [[domain]]. This "apportionment" can signify different things in various contexts, such as relative proportion of observations, or [[information]] regarding the residual uncertainty of its true value.
It is necessary to express probability as a density function 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 as a ''probability density function'', 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
Such a function leads to the definition of an associated [[cumulative distribution function]].
Such a function leads to the definition of an associated [[cumulative distribution function]].