# First Central Moment is Zero

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 $0$.
 $\ds \mu_1$ $=$ $\ds \expect {X - \mu}$ Definition of Central Moment $\ds$ $=$ $\ds \expect X - \mu$ Linearity of Expectation Function $\ds$ $=$ $\ds \mu - \mu$ $\expect X = \mu$ $\ds$ $=$ $\ds 0$
$\blacksquare$