Given two normal distribution random variables a,b;

Each has a mean of 0, a standard deviation of 1, and NO covariance between them;

Adding the two variables together will give a result with a standard deviation of (2)**0.5.

I am interested in a very similar problem;

Take only samples of a and b which have the same sign and add them; reject all samples in a and b which have opposite signs; compute the result.

The problem introduces a covariance that is positive, but of an unkown value.

I know experimentally, that the result will have a mean of 0, a standard deviation of ~1.809213(2),

However, I can't figure out the solution analytically.

Is this constant a known value that someone has computed before?

If not, can someone give me some pointers on how to calculate the constant?

I know how to take the integral of z*exp(-0.5*x**2)**2, and then using the area of the bell curve (2*pi)**0.5 to compute the variance or standard deviation of the distribution.

I'm just not sure how to set the problem up for two independent normal distributions that are added with the restrictions I have given...