Difference between Variance and 2nd moment

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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 $E(X^2) - E(X)^2$?

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2 Answers

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$$ \mathbb{E}(X^n) = \text{raw moment}\\ \mathbb{E}\left[\left(X-\mathbb{E}(X)\right)^n\right] = \text{central moment} $$ where the 2nd central moments represents the variance.

only equal when $\mathbb{E}(X) = 0$ as with $\mathcal{N}(0,1)$.

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Simple: $$\operatorname{Var}(X)\neq E(X^2)$$

The second moment is not, in general, equal to variance.

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