Moment¶

Definition¶

Let $X$ be a random variable and let $r\ge1$ be a natural number. If $\mathbb{E}|X|^r<\infty$$\infty$, the $r$th moment (or raw moment, moment about the origin) of $X$ is

Writing $\mu=\mathbb{E}X$ for the first moment, the $r$th central moment (moment about the mean) is

If $\mathbb{E}|X|^r=\infty$$\infty$, the $r$th moment does not exist. Both moments are expectations of a function of $X$, so by LOTUS they are computed from the PMF as $\displaystyle \sum_x x^r p_X(x)$ or from the PDF as $\int x^r f_X(x)\,\mathrm{d}x$, and similarly with $(x-\mu)^r$$\mu {)}^r$.

Interpretation¶

Moments summarize a distribution through averages of powers. Raw moments are measured from $0$; central moments are measured from the mean, so they describe the shape of the distribution around its centre and do not change when $X$ is shifted. Higher powers give large deviations more weight, so higher moments are increasingly sensitive to the tails.

For $X$ taking the values $1,2,3$ with probabilities $0.5,0.3,0.2$:

• raw moments: $\mathbb{E}X=1.7$$1.7$, $\mathbb{E}[X^2]=0.5+1.2+1.8=3.5$ and $\mathbb{E}[X^3]=0.5+2.4+5.4=8.3$$8.3$;
• central moments: the deviations from $\mu=1.7$ are $-0.7,0.3,1.3$, so the second central moment is $0.5(0.49)+0.3(0.09)+0.2(1.69)=0.61$ and the third is $0.5(-0.343)+0.3(0.027)+0.2(2.197)=0.276$$0.276$.

The positive third central moment reflects the long right tail: the single large deviation $1.3$ outweighs the more likely deviation $-0.7$ once it is cubed.

The order of operations matters. The second moment $\mathbb{E}[X^2]$ averages the squares, whereas $(\mathbb{E}X)^2$ squares the average; here $3.5$ against $1.7^2=2.89$$2.89$. Their difference is the second central moment, $3.5-2.89=0.61$$0.61$.

Key Results¶

The first moment is the expectation, and the first central moment is always $0$. The second central moment is the variance; expanding the power and using linearity of expectation expresses central moments through raw moments:

The cross terms do not cancel: the third central moment is not $\mathbb{E}[X^3]-\mu^3$${\mu }^3$. When the third moment exists and the standard deviation satisfies $\sigma>0$$0$, the third central moment divided by $\sigma^3$ is the skewness, a measure of asymmetry. This standardized coefficient is undefined for a constant random variable, for which $\sigma=0$$0$.

• Since a variance is never negative, $\mathbb{E}[X^2]\ge(\mathbb{E}X)^2$$(\mathbb{E}X{)}^2$, with equality exactly when $X$ is constant almost surely.
• For constants $a,b$, the $r$th central moment of $a+bX$ is $b^r$ (not $|b|^r$) times that of $X$. In particular, shifting by $a$ leaves all central moments unchanged, and the odd central moments change sign when $b=-1$$-1$. Raw moments are not shift-invariant: $\mathbb{E}[(a+bX)^2]=a^2+2ab\,\mathbb{E}X+b^2\,\mathbb{E}[X^2]$$b^2\mathbb{E}[X^2]$.
• If the $s$th moment exists, so does every $r$th moment with $r\le s$$s$, because $|x|^r\le1+|x|^s$$|x{|}^s$. The converse fails: moments can fail to exist for heavy-tailed distributions even when lower moments are finite.
• If $X$ is bounded, say $|X|\le c$$c$, every moment exists and $|\mathbb{E}[X^r]|\le c^r$$c^r$.
• For an indicator random variable $I_A$, $I_A^r=I_A$$I_A$, so every moment equals $\mathbb{P}(A)$.
• If the moment generating function is finite on an open interval around $0$, all moments exist and $\mathbb{E}[X^n]=M_X^{(n)}(0)$ (moments from the MGF).

Examples¶

• For a fair die, $\mathbb{E}X=21/6=3.5$ and $\mathbb{E}[X^2]=(1+4+9+16+25+36)/6=91/6$$91/6$, so $\operatorname{Var}(X)=91/6-49/4=35/12$$35/12$.
• For $X\sim\operatorname{Bern}(p)$ (Bernoulli distribution), $X^r=X$$X$, so $\mathbb{E}[X^r]=p$ for every $r$ and $\operatorname{Var}(X)=p-p^2=p(1-p)$$p)$.
• For $X\sim\operatorname{Unif}(0,1)$$\mathrm{Unif}(0,1)$, $\mathbb{E}[X^r]=\int_0^1 x^r\,\mathrm{d}x=1/(r+1)$$1)$, so $\operatorname{Var}(X)=1/3-1/4=1/12$$1/12$.
• For $X\sim\operatorname{Exp}(\lambda)$$\mathrm{Exp}(\lambda )$, $\mathbb{E}[X^n]=n!/\lambda^n$$n!/{\lambda }^n$; the second central moment is $2/\lambda^2-1/\lambda^2=1/\lambda^2$$1/{\lambda }^2$.
• For the three-point $X$ above and $Y=2X+1$$1$, $\mathbb{E}Y=2(1.7)+1=4.4$$4.4$, the second central moment of $Y$ is $2^2(0.61)=2.44$ and the third is $2^3(0.276)=2.208$$2.208$.
• If $X$ has a symmetric distribution about its mean, every odd central moment that exists is $0$: the deviations $X-\mu$ and $\mu-X$ have the same distribution.
• The density $f(x)=2x^{-3}$ for $x\ge1$ has finite mean $\mathbb{E}X=\int_1^\infty 2x^{-2}\,\mathrm{d}x=2$$2$, but $\mathbb{E}[X^2]=\int_1^\infty 2x^{-1}\,\mathrm{d}x=\infty$$\infty$, so its variance does not exist.

Prerequisites¶

• Expectation
• Random Variable
• LOTUS

Related Concepts¶

• Variance and Standard Deviation
• Moment Generating Function
• Moments from the MGF
• Symmetric Distribution
• Linearity of Expectation
• Bernoulli Distribution

Sources¶

• Nicky van Foreest · Probability Distributions · Week 1

Aliases: raw moment, moment about the origin, central moment, moment about the mean Keywords: nth moment, first moment, second moment, nth central moment, E[X^r], existence of moments Symbols: infinity, integral, lambda, mu, sigma, sum

Last updated: 27 October 2026 11:52