Changes

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]].