Moment

Definition

Let $X$ be a real random variable and let $k \ge 1$ be an integer with $E[|X|^k] < \infty$ . The $k$ -th moment of $X$ is $m_k = E[X^k].$ The $k$ -th absolute moment is $E[|X|^k]$ .

Remarks

By the law of the unconscious statistician, $m_k = \int x^k \, \mu_X(dx)$ . So the moments depend only on the law $\mu_X$ of $X$ . The first moment is the mean. The variance is $m_2 - m_1^2$ . If $E[|X|^k] < \infty$ , then $E[|X|^j] < \infty$ for each $j \le k$ . The Cauchy distribution has no moments of order $k \ge 1$ .

Tag	`0PH0VHTQS0`
URI	`m2://0PH0VHTQS0`
Kind	definition
Writing	W1 stated
MSC	60E05
Defines	Moment
Uses	Expected value, Law of the unconscious statistician

Tag

0PH0VHTQS0

URI

m2://0PH0VHTQS0

Kind

definition

Writing

W1 stated

MSC

60E05

Defines

Moment

Uses

Expected value, Law of the unconscious statistician

Moment

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

Used by