What is the correlation of a truncated bivariate Normal distribution? $X$ and $Y$ are independent standard normal variables.  If we generate $n$ samples $(x,y),$ what is the correlation for samples $(x, y)$ with  $x+y\gt0$?
 A: The question asks about the correlation coefficient of a truncated standard bivariate Normal distribution, where it is limited to the half-plane $X+Y\gt 0.$  (See the figure below.)
To work with this, it's convenient to let $U=X+Y$ (truncated to $U\gt 0$) and $V=X-Y$ (not truncated at all).  Evidently

*

*$U/\sqrt 2$ has a Half-normal distribution,


*$V/\sqrt 2$ has a standard Normal distribution, and


*$(U,V)$ is independent.
The last observation follows directly from the independence of $(X+Y, X-Y).$
It is immediate that

*

*$E[U/\sqrt 2] = \sqrt{2/\pi},$


*$E[V] = 0,$


*$E[U^2/2] = 1,$


*$E[V^2/2] = 1,$ and


*$E[UV] = E[U]E[V] = 0.$
Consequently


*$\operatorname{Var}(U) = E[U^2]-E[U]^2 = 2 - 2/\pi,$


*$\operatorname{Var}(V) = E[V^2]-E[V]^2 = 2,$ and


*$\operatorname{Cov}(U,V) = E[UV] - E[U]E[V] = 0.$
The question asks about the correlation of $((U+V)/2, (U-V)/2),$ whose formula is
$$\begin{aligned}
\rho = \operatorname{Cor}((U+V)/2, (U-V)/2) &= \operatorname{Cor}(U+V, U-V) \\ &=\frac{\operatorname{Cov}(U+V,U-V)} {\sqrt{\operatorname{Var}(U+V)\operatorname{Var}(U-V)}}.
\end{aligned}$$
The standard rules of covariance (namely, its bilinarity) reduce these expressions to


*$\operatorname{Cov}(U+V,U-V) = \operatorname{Var}(U)-  \operatorname{Var}(V) = -2/\pi,$

*$\operatorname{Var}(U\pm V) = \operatorname{Var}(U) + \operatorname{Var}(V) \pm 2\operatorname{Cov}(U,V) = 2-2/\pi.$
The formula for $\rho$ thereby simplifies to

$$\rho = \frac{-2/\pi}{\sqrt{(2 - 2/\pi)^2}} = \frac{\pi}{1-\pi} \approx -0.4669422.$$


This plot shows 2000 independent $(X,Y)$ values.  Those with $X+Y\gt 0$ are darker.  Their least-squares fit appears in red: it is a line expected to have slope $\pi/(1-\pi)$ passing through the point $(\sqrt{1/\pi}, \sqrt{1/\pi}).$  It comes very close to that in this sample, helping to check the correctness of the formulas.
