Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Further moments ▷ First standardized moment is zero

Theorem: The first standardized moment is zero, i.e.

\[\label{eq:momstand-1st} \mu_1^{*} = 0 \; .\]

Proof: The first standardized moment of a random variable $X$ with expected value $\mu$ and standard deviation $\sigma$ is defined as

\[\label{eq:momstand-1st-def} \mu_1^{*} = \frac{\mathrm{E}[(X-\mu)^1]}{\sigma^1} \; .\]

Due to the linearity of the expected value and by plugging in $\mu = \mathrm{E}(X)$, we have

\[\label{eq:momstand-1st-qed} \begin{split} \mu_1^{*} &= \frac{\mathrm{E}[X-\mu]}{\sigma} \\ &= \frac{\mathrm{E}(X) - \mu}{\sigma} \\ &= \frac{\mathrm{E}(X) - \mathrm{E}(X)}{\sigma} \\ &= 0 \; . \end{split}\]
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Metadata: ID: P526 | shortcut: momstand-1st | author: JoramSoch | date: 2026-02-27, 11:43.