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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

 

|[[Image:Uniform.png|px=133]]

The '''probability density function''' of a continuous [[random variable]] is the function that provides the likelihood that the variable will have a value in a given interval when the function is integrated over that same interval.  Stated another way, a '''probability density function''' ''f'' is a non-negative valued real 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>

:<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 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.  

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]].

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:

In order to qualify as a ''probability density function'', such a function must satisfy the following two criteria:

 

(1) <math> f(x) \geq 0 </math>  <math> \forall x </math>  inside the domain of support.

(1) <math> f(x) \geq 0 </math>  <math> \forall x </math>  inside the domain of support.

(2) <math> \int_{-\infty}^\infty \,f(x)\,dx = 1. </math>  i.e., finitely convergent (to unity by convention).

(2) <math> \int_{-\infty}^\infty \,f(x)\,dx = 1. </math>  i.e., finitely convergent (to unity by convention).

(3) <math> \int_{-\infty}^a \,f(x)\,dx \leq \int_{-\infty}^b \,f(x)\,dx </math>  for a<b,  i.e., is non-decreasing

<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]].

[[Category:mathematics]]

[[Category:Probability and Statistics]]