Proof: First central moment is zero
Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Further moments ▷ First central moment is zero
Metadata: ID: P167 | shortcut: momcent-1st | author: JoramSoch | date: 2020-09-09, 07:51.
Theorem: The first central moment is zero, i.e.
\[\label{eq:momcent-1st} \mu_1 = 0 \; .\]Proof: The first central moment of a random variable $X$ with mean $\mu$ is defined as
\[\label{eq:momcent-1st-def} \mu_1 = \mathrm{E}\left[ (X-\mu)^1 \right] \; .\]Due to the linearity of the expected value and by plugging in $\mu = \mathrm{E}(X)$, we have
\[\label{eq:momcent-1st-qed} \begin{split} \mu_1 &= \mathrm{E}\left[ X-\mu \right] \\ &= \mathrm{E}(X) - \mu \\ &= \mathrm{E}(X) - \mathrm{E}(X) \\ &= 0 \; . \end{split}\]∎
Sources: - ProofWiki (2020): "First Central Moment is Zero"; in: ProofWiki, retrieved on 2020-09-09; URL: https://proofwiki.org/wiki/First_Central_Moment_is_Zero.
Metadata: ID: P167 | shortcut: momcent-1st | author: JoramSoch | date: 2020-09-09, 07:51.