Does a second moment of zero imply a mean of zero?

Question

I was wondering about this. I have a second moment that is zero. Can I conclude that if $E[x^2]=0$ then $Var[x]=E[x^2]-E[x]E[x]$ implies $E[x]=0$ since the variance must be non-negative? (If not, can I imply anything else from that, in particular, regarding if we also have mean square convergence?)

Answer 1

The answer to your question is yes.

Alternatively, you can use Hölder's inequality and obtain the following inequality $$ 0\le|\operatorname EX|\le\operatorname E|X|\le(\operatorname E|X|^2)^{1/2}. $$