Difference Between Variance and 2Nd Moment
$\Begingroup$ I Understand That $Var(X) = E(X^2) - E(X)^2 $ and That the Second Moment, Variance, Is $E(X^2)$ How Is Variance Simultaneously $E(X^2)$ and...
I understand that
$Var(X) = E(X^2) - E(X)^2 $
And that the second moment, variance, is
$E(X^2)$
How is variance simultaneously $E(X^2)$ and $E(X^2) - E(X)^2$?
2 Answers
$$ \mathbb{E}(X^n) = \text{raw moment}\\ \mathbb{E}\left[\left(X-\mathbb{E}(X)\right)^n\right] = \text{central moment} $$ where the 2nd central moments represents the variance.
only equal when $\mathbb{E}(X) = 0$ as with $\mathcal{N}(0,1)$.
Simple: $$\operatorname{Var}(X)\neq E(X^2)$$
The second moment is not, in general, equal to variance.
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