Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Further moments ▷ Second raw moment as sum of squares

Theorem: The second raw moment can be expressed as the sum of squared expected value and squared standard deviation, i.e.

\[\label{eq:momraw-2nd} \mu_2' = \mathrm{E}(X)^2 + \sigma(X)^2 \; .\]

Proof: The second raw moment of a random variable $X$ is defined as

\[\label{eq:momraw-2nd-def} \mu_2' = \mathrm{E}\left[ (X-0)^2 \right] \; .\]

Using the partition of variance into expected values

\[\label{eq:var-mean} \mathrm{Var}(X) = \mathrm{E}(X^2) - \mathrm{E}(X)^2\]

and the relationship between standard deviation and variance

\[\label{eq:std-var} \sigma(X) = \sqrt{\mathrm{Var}(X)} \; ,\]

the second raw moment can be rearranged into:

\[\label{eq:momraw-2nd-qed} \begin{split} \mu_2' &\overset{\eqref{eq:momraw-2nd-def}}{=} \mathrm{E}(X^2) \\ &\overset{\eqref{eq:var-mean}}{=} \mathrm{Var}(X) + \mathrm{E}(X)^2 \\ &= \mathrm{E}(X)^2 + \sigma(X)^2 \; . \end{split}\]
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Metadata: ID: P172 | shortcut: momraw-2nd | author: JoramSoch | date: 2020-10-08, 05:05.