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