# Expected value of squared random variable if expected value of random variable is $0$

Let

$E[X]=0$

them How is Expected value of $E[X^2]$ ? in my opinion is $0$ But I need confirmation and explanation

In general, if $(\Omega,\Sigma,P)$ is a probability space and $X: (\Omega,\Sigma) \to (\mathbb{R},\mathcal{B}(\mathbb{R}))$ is a real-valued random variable, then $$\text{E}[X^{2}] = \int_{\Omega} X^{2} ~ d{P}.$$

see for more about this in this quetion Computing the Expectation of the Square of a Random Variable: $E[X^2]$.

• If X is 1 half the time and -1 half the time, E(X) =0, bur X^2=1 all the time, so E(X^2)=1. Hope this example helps see whats going on Sep 10, 2021 at 16:15

No way. Since $X^2\geq 0$, $\mathbb{E}[X^2]=0$ iff $X$ is zero almost surely.
Otherwise, if $\mathbb{E}[X^2]$ is well defined, $\mathbb{E}[X^2]\color{red}{>}0.$
• You can't say anything about $E(X^2)=V(X)$. For example, suppose $X\sim N(0,a)$ where $a>0$ is fixed and arbitrary. Then $E(X)=0$ whereas $E(X^2)=a$. Jan 7, 2021 at 1:41
A really easy example to disabuse yourself of this notion is to look up standard normal and $$\chi^2$$ distributions.
If $$Z \sim N(0, 1)$$, then $$Z^2 \sim \chi_1^2$$. We have $$\mathrm E[Z] = 0$$, but $$\mathrm E[Z^2] = 1$$.