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## Homework Statement

Have 2 iid random variables following the distribution [tex] f(x) = \frac{\lambda}{2}e^{-\lambda |x|}, x \in\mathbb{R}[/tex]

I'm asked to solve for [tex] E[X_1 + X_2 | X_1 < X_2] [/tex]

## Homework Equations

## The Attempt at a Solution

So what I'm trying to do is create a new random variable[tex] Z = X_1 + X_2 [/tex] When I do this I get the following convolution formula for its density[tex] g(z) = \int_{-\infty}^{\infty} \frac{\lambda^2}{4} e^{-\lambda |z- x_1|} e^{-\lambda |x_1|} dx_1[/tex]

I'd really only like some advice on how to go about attacking this integral. It looks to me like I need to break it down into cases depending on z<x1 or z>x1 but that doesn't seem like it will produce a clean solution to me.

Or if you can see that I'm attacking this problem completely the wrong way and I shouldn't even be trying to do this please let me know. No alternative method of attack needed. I can try to figure out other ways if this is completely off base.

Thanks

**edit:**

I've had a thought. If X1 > 0 then [tex] Z = X_1 + X_2 \gt 2X_1 \gt X_1[/tex] So now if I can do somthing similar for X1 < 0 I can evaluate the integral.

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