# Difference between Variance and 2nd moment

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

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

• Under which conditions is the second moment equal to variance? Nov 25 '16 at 17:32
• @slimydummy If $E(X^2)-(E(X))^2=E(X^2)$, so use some algebra and you get...
– user223391
Nov 25 '16 at 17:33
• That was dumb of me. Thank you Nov 25 '16 at 17:35
• @slimydummy True to your name. :) But the thruth is that there are no stupid question but maybe stupid answers. Nov 25 '16 at 17:51