Expectations of normal variables

[lequan]

\begin{eqnarray*}
&&\mathbb{E}\left( 1_{\left\{ X_{1}+X_{2}>\rho \right\} }X_{1}X_{2}\right)
\\
&&\mathbb{E}\left( 1_{\left\{ X_{1}+X_{2}>\rho \right\} }X_{1}^{2}\right)
\end{eqnarray*}

where $X_1$ and $X_2$ are independent normal variables. I am wondering whether there exist closed-form expressions for the above two expectations.

 

[andrewkirk]

What would you consider to be a 'closed-form expression'?
With a fairly common, narrow interpretation of 'closed form', even a simple univariate normal calculation such as $Pr(X_1<x)$ does not have a closed form because there is no analytic formula for the indefinite integral of the normal pdf.
Conversely, taking a broader interpretation, the two expressions you have written above could be considered to already be 'closed-form'.