Central moment

Definition

Let $X$ be a real random variable and let $k \ge 1$ be an integer with $E[|X|^k] < \infty$ . Let $\mu = E[X]$ . The $k$ -th central moment of $X$ is $\mu_k = E\left[(X - \mu)^k\right].$

Remarks

Always $\mu_1 = 0$ , and $\mu_2 = \operatorname{Var}(X)$ is the variance. Central moments do not change when one adds a constant to $X$ . For $a \in \mathbb{R}$ , the central moments of $aX$ are $a^k \mu_k$ . The binomial theorem gives $\mu_k = \sum_{j=0}^{k} \binom{k}{j} (-\mu)^{k - j} m_j$ in terms of the moments $m_j = E[X^j]$ , with $m_0 = 1$ . For a distribution that is symmetric about its mean, all odd central moments are $0$ .

Tag	`0ZT24EC90K`
URI	`m2://0ZT24EC90K`
Kind	definition
Writing	W1 stated
MSC	60E05
Defines	Central moment
Uses	Moment, Expected value, Variance

Tag

0ZT24EC90K

URI

m2://0ZT24EC90K

Kind

definition

Writing

W1 stated

MSC

60E05

Defines

Central moment

Uses

Moment, Expected value, Variance

Central moment

Definition

Remarks

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