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 $.