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.

Answer

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

\begin{align}
P(XY>0)&=P(X>0,Y>0)+P(X<0,Y<0)
\\&=P(X>0,Y>0)+P(-X>0,-Y>0)
\end{align}

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

Similarly,

\begin{align}
P(XY<0)&=P(X>0,Y<0)+P(X<0,Y>0)
\\&=P(X>0,-Y>0)+P(-X>0,Y>0)
\end{align}

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

P(XY<0)=2P(X>0,-Y>0)

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

P(X>0,-Y>0)=\frac14-\frac1{2\pi}\sin^{-1}\rho

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

Leave a Comment