# Second Standardized Moment is One

Let $X$ be a random variable on some probability space with standard deviation $\sigma$.
Then the second standardized moment $\alpha_2$ of $X$ is equal to $1$.
 $\ds \alpha_2$ $=$ $\ds \frac {\mu_2} {\sigma^2}$ Definition of Standardized Moment $\ds$ $=$ $\ds \frac {\sigma^2} {\sigma^2}$ Definition of Central Moment $\ds$ $=$ $\ds 1$
$\blacksquare$