$\alpha, \alpha', \beta$ and $\beta'$ are four independent standard normal random variables, I am wondering how to compute the probability of the following two events:

  • $\alpha>\alpha'>0, \ \ \beta<\beta', \ \ |\alpha-\alpha'+\beta-\beta'|\geq c_1,$
  • $\alpha>0>\alpha', \ \ \beta<\beta', \ \ |\alpha-\alpha'+\beta-\beta'|\geq c_1, \ \ |\frac{\alpha}{\alpha'}|>c_2$

where $c_1$ and $c_2$ are positive constants. I know that the sum and difference of normal random variables are still normal random variables, but I'm not sure how to use this with the inequality relation here.

  • $\begingroup$ These can be calculated with four-dimensional integrals, but they have no simple formulas. $\endgroup$
    – Matt F.
    Sep 3, 2020 at 16:37
  • $\begingroup$ Even the probabilities for simpler events, like $\alpha>0, \ \alpha+\beta>c_1$ or $\alpha-\alpha'>c_1,\ \alpha/\alpha'>c_2$ do not have simple formulas in terms of the usual functions for normal variables. $\endgroup$
    – Matt F.
    Sep 3, 2020 at 17:25
  • $\begingroup$ @MattF. Thank you for your answer! Is it possible to simplify the formula so that we can have a lower bound of the probability? $\endgroup$
    – luw
    Sep 3, 2020 at 22:56
  • $\begingroup$ what are you doing with these probabilities? There is trivially "a lower bound", so the question has to be whether there's a useful lower bound, and that will depend on your use for the probabilities. $\endgroup$
    – Matt F.
    Sep 4, 2020 at 0:55
  • $\begingroup$ @MattF. I am analyzing some machine learning algorithms and I want to say that parameters initialized as standard normal variables have the properties above with certain probability. So I guess any lower bound would be good for me. $\endgroup$
    – luw
    Sep 4, 2020 at 3:52

1 Answer 1


This might work, the pdf might have the answer or at least part of the question but could not find the rest. https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter4.pdf


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