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
\(\ds \) \(=\) \(\ds \expect X - \mu\) Linearity of Expectation Function
\(\ds \) \(=\) \(\ds \mu - \mu\) $\expect X = \mu$
\(\ds \) \(=\) \(\ds 0\)

$\blacksquare$