The Question
We are given a random variable $X$ on a probability space $(\Omega, F, P)$. $X \sim N(0,\sigma^2)$. Given an event $H \in F$, what is the largest possible conditional expectation $E[X\vert H]$. This bound should be given in terms of $P(H)$.
What I've Done
I reason that, worst case, $H=\{X>\lambda\}$ for some $\lambda$. So I compute \begin{align*} E[X\vert H] &\leq \frac{1}{P(\{X>\lambda\})}\int_{\lambda}^\infty \frac{x}{\sqrt{2\pi\sigma^2}}e^{\frac{-x^2}{2\sigma^2}}dx \\ &=\frac{\frac{\sigma}{\sqrt{2\pi}}e^{\frac{-\lambda^2}{2\sigma^2}}}{\int_{\lambda}^\infty \frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-x^2}{2\sigma^2}}dx} \\ &=\frac{\sigma^2e^{\frac{-\lambda^2}{2\sigma^2}}}{\int_{\lambda}^\infty e^{\frac{-x^2}{2\sigma^2}}dx} \end{align*}
From here I'm thinking I could get this in to some manageable function of $\lambda$ or perhaps an approximation, and somehow convert that into a function $P(H)$.
I guess, ultimately it seems to come down to finding $\int_{\lambda}^\infty \frac{x}{\sqrt{2\pi\sigma^2}}e^{\frac{-x^2}{2\sigma^2}}dx$ as a function of $P(\{X>\lambda\})$. One observation I have is that the former appears to be the derivative of the latter (up to a constant). Not sure if that helps.
I'm not too knowledgeable about probability, so perhaps there is a much simpler or better approach altogether. Any help would be greatly appreciated.