Probability density function

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 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).

(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.

If the domain of the variable is finite, then the infinite limits on the above integrals would be replaced by those bounds, and a third requirement would be:

(3) <math> f(x) = 0 </math> <math> \forall x </math> outside the finite domain of support.