Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Further moments ▷ Second raw moment and variance

Theorem: The second raw moment can be expressed as

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

where $\mathrm{Var}(X)$ is the variance of $X$ and $\mathrm{E}(X)$ is the expected value of $X$.

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

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

the second raw moment can be rearranged into:

\[\label{eq:momraw-2nd-qed} \mu_2' \overset{\eqref{eq:momraw-2nd-def}}{=} \mathrm{E}(X^2) \overset{\eqref{eq:var-mean}}{=} \mathrm{Var}(X) + \mathrm{E}(X)^2 \; .\]
∎
Sources:

Metadata: ID: P172 | shortcut: momraw-2nd | author: JoramSoch | date: 2020-10-08, 05:05.