Expectation of a random variable squared

Question

What is the reason that (why) $$\mathbb{E}\left[ X \right]=0,\, \operatorname{var}\left[ X \right]=1 \Leftrightarrow \mathbb{E}\left[ X^{2} \right]=1$$

Answer 1

Use the fact that $\color{blue}{\operatorname{var}(X) = \Bbb{E}\left[X^2\right] - \left(\Bbb{E}[X]\right)^2}$. So if $\Bbb{E}[X]=0$, then $\operatorname{var}(X) = \Bbb{E}\left[X^2\right] $.

Answer 2

The variance of $X$ is by definition the expectation of $(X-\mathbb EX)^2$.

So if $\mathbb EX=0$ then it is the expectation of $(X-0)^2=X^2$.