First Central Moment is Zero
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Theorem
Let $X$ be a random variable on some probability space with mean $\mu$.
Then the first central moment $\mu_1$ of $X$ is equal to $0$.
Proof
\(\ds \mu_1\) | \(=\) | \(\ds \expect {X - \mu}\) | Definition of Central Moment | |||||||||||
\(\ds \) | \(=\) | \(\ds \expect X - \mu\) | Expectation is Linear | |||||||||||
\(\ds \) | \(=\) | \(\ds \mu - \mu\) | $\expect X = \mu$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
$\blacksquare$