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

$P(a\leq x\leq b)=\int _{a}^{b}f(x)\,dx$

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) $\int _{-\infty }^{\infty }\,f(x)\,dx=1.$ i.e., finitely convergent (to unity by convention).

(2) $\int _{-\infty }^{a}\,f(x)\,dx\leq \int _{-\infty }^{b}\,f(x)\,dx$ for a<b, i.e., is non-decreasing