Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Further moments ▷ First raw moment is mean

Theorem: The first raw moment equals the mean, i.e.

\[\label{eq:momraw-1st} \mu_1' = \mu \; .\]

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

\[\label{eq:momraw-1st-def} \mu_1' = \mathrm{E}\left[ (X-0)^1 \right]\]

which is equal to the expected value of $X$:

\[\label{eq:momraw-1st-qed} \mu_1' = \mathrm{E}\left[ X \right] = \mu \; .\]
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Metadata: ID: P171 | shortcut: momraw-1st | author: JoramSoch | date: 2020-10-08, 04:19.