# Absolute moment

of a random variable $X$

The mathematical expectation of $| X | ^ {r}$, $r > 0$. It is usually denoted by $\beta _ {r}$, so that

$$\beta _ {r} = {\mathsf E} | X | ^ {r} .$$

The number $r$ is called the order of the absolute moment. If $F (x)$ is the distribution function of $X$, then

$$\tag{1 } \beta _ {r} = \int\limits _ {- \infty } ^ { {+ } \infty } | x | ^ {r} d F ( x ) ,$$

and, for example, if the distribution of $X$ has density $p (x)$, one has

$$\tag{2 } \beta _ {r} = \int\limits _ {- \infty } ^ { {+ } \infty } | x | ^ {r} p ( x ) dx .$$

In relation to the equations (1) and (2) one also speaks, respectively, of the absolute moments of the distribution function $F(x)$ and the density $p(x)$. The existence of $\beta _ {r}$ implies the existence of the absolute moment $\beta _ {r ^ \prime }$ and also of the moments (cf. Moment) of order $r ^ \prime$, for $0 < r ^ \prime \leq r$. Absolute moments often appear in estimates of probability distributions and their characteristic functions (cf. Chebyshev inequality in probability theory; Lyapunov theorem). The function $\mathop{\rm log} \beta _ {r}$ is a convex function of $r$, and the function $\beta _ {r} ^ {1/r}$ is a non-decreasing function of $r$, $r > 0$.

How to Cite This Entry:
Absolute moment. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absolute_moment&oldid=45005
This article was adapted from an original article by Yu.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article