Second Standardized Moment is One

Theorem

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 $1$.

Proof

\(\ds \alpha_2\)

\(=\)

\(\ds \frac {\mu_2} {\sigma^2}\)

Definition of Standardized Moment

\(\ds \)

\(=\)

\(\ds \frac {\sigma^2} {\sigma^2}\)

Definition of Central Moment

\(\ds \)

\(=\)

\(\ds 1\)

$\blacksquare$