Can I figure out $E(X)$ from $E(X^2)$?

Question

I have an expression for $E(X^2)$, how can I get $E(X)$?

Answer 1

Consider the distributions given by $P_1(x) = \frac{2}{\sqrt{\pi}}e^{-x^2}$ on $(0, \infty)$ and $P_2(x) = \frac{1}{\sqrt{\pi}}e^{-x^2}$ on $(-\infty,\infty)$. Both have second moment $E[X^2] = \frac{1}{2}$, but the mean of $P_1$ is positive since it is defined on $(0,\infty)$ and the mean of $P_2$ is $0$.

If you know $E[X^2]$ and $\operatorname{Var}(X) = E[X^2] - E[X]^2$, you have that $$|E[X]| = \sqrt{E[X^2] - \operatorname{Var}(X)}$$ so if you have some information on the random variable (for example, if it was strictly positive) then you can determine $E[X]$.