Proof: Second central moment is variance
Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Further moments ▷ Second central moment is variance
Metadata: ID: P173 | shortcut: momcent-2nd | author: JoramSoch | date: 2020-10-08, 05:13.
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:
\[\label{eq:momraw-1st-qed} \mu_2 = \mathrm{E}\left[ (X - \mathrm{E}(X))^2 \right] = \mathrm{Var}(X) \; .\]∎
Sources: - Wikipedia (2020): "Moment (mathematics)"; in: Wikipedia, the free encyclopedia, retrieved on 2020-10-08; URL: https://en.wikipedia.org/wiki/Moment_(mathematics)#Significance_of_the_moments.
Metadata: ID: P173 | shortcut: momcent-2nd | author: JoramSoch | date: 2020-10-08, 05:13.