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 its entire domain. 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