Moment ( $E [X^{n}]$ )

A moment integrates a power of a quantity against a specified measure. For a real random variable

X

, the

n

E [X^{n}]

when defined; the

n

th central moment instead measures powers of deviation from its mean. For equally likely values one and three, the first raw moment is two and the second raw moment is five, while the second central moment is one, its variance. Moments summarize features of a law and can be easier to compute than its full probability distribution. Even knowing every moment does not always determine the law: that additional property is moment determinacy.

Raw moment ( $E [X^{r}]$ )

The

r

X

E [X^{r}]

, when it exists. A central moment instead uses

E [(X - E [X])^{r}]

E [X^{4}]

. Its finiteness is equivalent to square integrability of

X^{2}

. For centered variables, dividing it by the squared variance gives kurtosis.

E [X^{2}]

. Its variance is

E [X^{2}] - (E X)^{2}

when these quantities are finite.

p > 0

X

E [∣ X ∣^{p}]

when this expectation is finite.

μ

, its

n

E [(X - μ)^{n}]

when defined. The first is zero; the second is the variance; the standardized third gives skewness. Translating

X

by a constant leaves central moments unchanged, because the mean translates by the same amount. For equally likely values one and three, the second central moment is one, whereas the second raw moment is five. Odd central moments can cancel between positive and negative deviations, so an absolute moment provides a different integrability check.

For a real random variable with finite third absolute moment and positive variance

σ^{2}

, its skewness is

γ_{1} = E [(X - E X)^{3}] / σ^{3}

. It is dimensionless and unchanged by positive rescaling or translation; a reflection reverses its sign. A symmetric distribution has zero skewness, but the converse fails: values