$X$ and $Y$ are independent standard normal variables. If we generate $n$ samples $(x,y),$ what is the correlation for samples $(x, y)$ with $x+y\gt0$?

  • 2
    $\begingroup$ The generation as such has no impact on the correlation. The pair $(X,Y)$ is distributed from a bivariate $N(0_2,I_2)$ distribution restricted to the half space $x+y>0$ and the correlation is defined as usual by$$\int_{x+y>0} (xy-\mathbb E[X]^2) c \exp\{-(x^2+y^2)/2\}\,\text dx\text dy$$where $c$ is the proper normalising constant. $\endgroup$
    – Xi'an
    Nov 17, 2021 at 7:05
  • $\begingroup$ thanks, so it means that the correlation still be zero? $\endgroup$
    – qwerty1010
    Nov 17, 2021 at 7:34
  • $\begingroup$ You have to compute this integral to check whether or not it is zero. $\endgroup$
    – Xi'an
    Nov 17, 2021 at 7:50
  • 1
    $\begingroup$ A plot of this truncated density strongly suggests the correlation ought to be negative, not zero. $\endgroup$
    – whuber
    Nov 17, 2021 at 15:07
  • $\begingroup$ You can easily approximate this quantity by simulationx=rnorm(T);y=rnorm(T);mean(x[x+y>0]*y[x+y>0])-mean(x[x+y>0])^2 $\endgroup$
    – Xi'an
    Nov 17, 2021 at 15:17

1 Answer 1


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

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

  2. $E[V] = 0,$

  3. $E[U^2/2] = 1,$

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

  5. $E[UV] = E[U]E[V] = 0.$


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

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

  3. $\operatorname{Cov}(U,V) = E[UV] - E[U]E[V] = 0.$

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

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

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.


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