Absolute moment

of a random variable

The mathematical expectation of , . It is usually denoted by , so that

The number is called the order of the absolute moment. If is the distribution function of , then

(1)

and, for example, if the distribution of has density , one has

(2)

In relation to the equations (1) and (2) one also speaks, respectively, of the absolute moments of the distribution function and the density . The existence of implies the existence of the absolute moment and also of the moments (cf. Moment) of order , for . Absolute moments often appear in estimates of probability distributions and their characteristic functions (cf. Chebyshev inequality in probability theory; Lyapunov theorem). The function is a convex function of , and the function is a non-decreasing function of , .