Sign of product of standard normal random variables

The question is mainly in the title:

Given two standard normal random variables with correlation \rho, what is the distribution of sign of their product?

I understand that when \rho=0, we have two iid standard normal random variables and therefore, they take positive and negative values independently with probability \frac12.

But I don’t know what to do if \rho\ne0. We can take \rho>0 without loss of generality (because if X and Y are std normal with correlation \rho, then -X and Y are std normal with correlation -\rho). But I could not proceed further.


As long as (X,Y) is standard bivariate normal with correlation \rho, the probability that XY is positive or negative can be found using the well-known result for the positive quadrant probability P(X>0,Y>0)=\frac14+\frac1{2\pi}\sin^{-1}\rho \tag{1}

(This is likely discussed here before but I cannot quite find the question.)

You have


Because (-X,-Y) has the same distribution as (X,Y), this probability is just P(XY>0)=2P(X>0,Y>0)



Again, (X,-Y) and (-X,Y) have the same distribution, so


And since (X,-Y) is bivariate normal with correlation -\rho, we have from (1) that


Source : Link , Question Author : Martund , Answer Author : StubbornAtom

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