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

  • $\begingroup$ MSE is a right forum for such type questions. Hint: integrate the density of the binormal distribution with zero correlation over the annular sector. $\endgroup$
    – user64494
    Dec 10, 2020 at 9:47
  • 1
    $\begingroup$ 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. $\endgroup$ Dec 10, 2020 at 12:43
  • $\begingroup$ @CarloBeenakker: Numerically for concrete values of $c_1,\,c_2$. $\endgroup$
    – user64494
    Dec 10, 2020 at 19:02


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