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A Expectations of normal variables

  1. Mar 5, 2016 #1
    \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.
     
  2. jcsd
  3. Mar 5, 2016 #2

    andrewkirk

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    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'.
     
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