Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Further moments ▷ First central moment is zero

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

$\label{eq:momcent-1st-qed} \begin{split} \mu_1 &= \mathrm{E}\left[ X-\mu \right] \\ &= \mathrm{E}(X) - \mu \\ &= \mathrm{E}(X) - \mathrm{E}(X) \\ &= 0 \; . \end{split}$
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Metadata: ID: P167 | shortcut: momcent-1st | author: JoramSoch | date: 2020-09-09, 07:51.