How to compute the following probability involving 4 normal random variables?

$$\alpha, \alpha', \beta$$ 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.

• These can be calculated with four-dimensional integrals, but they have no simple formulas. Sep 3 '20 at 16:37
• 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. Sep 3 '20 at 17:25
• @MattF. Thank you for your answer! Is it possible to simplify the formula so that we can have a lower bound of the probability?
– luw
Sep 3 '20 at 22:56
• 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. Sep 4 '20 at 0:55
• @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.
– luw
Sep 4 '20 at 3:52