Find $E[W|X>Y]$ where $W = X+Y$ and $X,Y \sim \exp(2)$ independently

Question

I need an idea on how to solve following conditional expectation $E[W|X>Y]$ where $W = X+Y$ and $X,Y \sim \exp(2)$ and $X$ and $Y$ are independent.

Thanks.

Answer 1

$$1=\mathbb{E}W=$$$$\mathbb{E}\left(W\mid X>Y\right)P\left(X>Y\right)+\mathbb{E}\left(W\mid X=Y\right)P\left(X=Y\right)+\mathbb{E}\left(W\mid X<Y\right)P\left(X<Y\right)=$$$$\mathbb{E}\left(W\mid X>Y\right)\frac{1}{2}+\mathbb{E}\left(W\mid X=Y\right)0+\mathbb{E}\left(W\mid X<Y\right)\frac{1}{2}=$$$$\mathbb{E}\left(W\mid X>Y\right)$$

Here symmetry is exploited: $\mathbb{E}\left(W\mid X>Y\right)=\mathbb{E}\left(W\mid X<Y\right)$ and $P(X>Y)=P(X<Y)$.