# Expectation over Gaussian random variable with mean zero and finite variance

Can someone explain me how is the expectation of a random variable (say $X$) calculated over Gaussian random variable with mean zero and a finite variance $\varepsilon\sim\mathcal{N}(0,\sigma^{2})$ ?. And what does it intuitively mean to calculate expectation of a random variable over a different random variable $E_\varepsilon(X)$, which in this case is $\varepsilon\sim\mathcal{N}(0,\sigma^2)$ ?

• What is $X$? Presumably some expression involving $\epsilon$ – Henry Oct 30 '17 at 12:44
• Suppose $X$ and $\epsilon$ are independent but we do know the distribution of X and lets assume it is a gaussian too ? –  redenzione11 Oct 30 '17 at 12:56
• If they are independent, then knowing things about $\epsilon$ tells you noting about $X$, and so the expectation of $X$ is what it was before $\epsilon$ was mentioned. This may not be what was intended – Henry Oct 30 '17 at 14:51
• You question is unclear. Are you asking about the conditional expectation $E(X|\epsilon)$ or integrating with respect to the measure associated to a normal variable? – Nap D. Lover Oct 30 '17 at 14:52
• @Henry I guess that would be the case with conditional expectation but I am talking about integrating with respect to the measure associated to a normal variable. –  redenzione11 Oct 30 '17 at 17:17