Conditional Covariance of X and Y given X+Y>0, X and Y are IID standard normal

Question

Question: Given that $X$ and $Y$ are standard normal IID, find $Cov(XY|X+Y>0)$.

I have tried the following: $$Cov(XY|X+Y>0) = E(XY|X+Y>0)-E(X|X+Y>0)\cdot E(Y|X+Y>0).$$ And $E(X|X+Y>0) = 1/\sqrt{\pi}$ since $E(X+Y|X+Y>0) = 2/\sqrt{\pi}$ using half-normal distribution or integrating directly.

I am not sure how to find the value of $E(XY|X+Y>0)$.

Answer 1

By definition: $$\mathbb{E}\left[XY\mid X+Y>0\right]P\left(X+Y>0\right)=\mathbb{E}XY1_{\{X+Y>0\}}$$ and in your situation $P\left(X+Y>0\right)$ is easy to find.

So it remains to find $\mathbb{E}XY1_{\{X+Y>0\}}$.

Define $U=X+Y$ and $V=X-Y$ .

Then it is not difficult to prove that $U$ and $V$ are independent and: $$XY1_{\{X+Y>0\}}=\frac{1}{4}\left(U^{2}-V^{2}\right)1_{\{U>0\}}$$

I leave the rest to you.