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
,
.
Absolute moment. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absolute_moment&oldid=17386