Difference between Variance and 2nd moment

$$ \mathbb{E}(X^n) = \text{raw moment}\\ \mathbb{E}\left[\left(X-\mathbb{E}(X)\right)^n\right] = \text{central moment} $$ 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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Variance

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