# How to compute the following probability involving two normal random variables?

$$\alpha$$ and $$\alpha'$$ are two independent standard normal random variables. What's the conditional probability $$\mathbb{P}[\alpha >0, \alpha' >0|c_1<|\alpha - \alpha'| where $$c_1$$ and $$c_2$$ are two positive constants.

• MSE is a right forum for such type questions. Hint: integrate the density of the binormal distribution with zero correlation over the annular sector. Dec 10 '20 at 9:47
• I don't see a closed form answer arriving; you would need to be able to find the indefinite integral of $e^{-(x-a)^2}\,{\rm erf}(x)$ , which I don't think has a closed form expression. Dec 10 '20 at 12:43
• @CarloBeenakker: Numerically for concrete values of $c_1,\,c_2$. Dec 10 '20 at 19:02