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

Theorem: The second central moment equals the variance, i.e.

\[\label{eq:momcent-2nd} \mu_2 = \mathrm{Var}(X) \; .\]

Proof: The second central moment of a random variable $X$ with mean $\mu$ is defined as

\[\label{eq:momcent-2nd-def} \mu_2 = \mathrm{E}\left[ (X-\mu)^2 \right]\]

which is equivalent to the definition of the variance:

\[\label{eq:momraw-1st-qed} \mu_2 = \mathrm{E}\left[ (X - \mathrm{E}(X))^2 \right] = \mathrm{Var}(X) \; .\]

Metadata: ID: P173 | shortcut: momcent-2nd | author: JoramSoch | date: 2020-10-08, 05:13.