Second Standardized Moment is One
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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 | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
$\blacksquare$