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Given $x \sim \mathcal{N}(x|0;1)$ and $y \sim \mathcal{N}(y|1;1)$,$x,y$ are two independent variables. How to find the expectation $\int \int (1+|x-y| )e^{-|x-y|}\mathcal{N}(x|0;1)\mathcal{N}(y|1;1) dx dy$. In other words, how to find the expecation of $f=(1+|x-y| )e^{-|x-y|}$ when $x,y$ are Gaussian random variables.

The absolute value makes the problem somewhat difficult.

Thank you so much

Phong Le
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  • Are $x,y$ independent, or do you know something else about their joint distribution? Do you have any reason to think there should be a nice closed form for this expectation? – Nate Eldredge Feb 18 '15 at 01:51
  • Thanks for your comments. x,y are two independent Gaussian random variables. I already edited my post. I hope that we can compute a closed form of this expectation, so it would be very helpful for my work later. – Phong Le Feb 18 '15 at 02:37

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