For any 3 independent random variables X1, X2 and X3 drawn from Gaussians with means μ1, μ2 and μ3, and standard deviations σ1, σ2 and σ3. I want to find a variety of joint and conditional probabilities. For example:
1.– the probability that X1 is greater than both X2 and X3.
2.– the probability that X3 is greater than X2 given than X1 is not the highest of the three (that is either X1 < X2 and/or X1 < X3).
I have run some simulations and realised that the even when the variables are drawn independently, the pairwise differences are not independent. However, I haven't been able to figure out a way to compute precisely these probabilities. Simulating the conditional probabilities described in my second point is computationally prohibitive when conditioning on cases where X1 is not the highest of the three but the mean of the distribution that it is drawn from (μ1) is much higher than the mean of the other two distributions (μ2 and μ3).
Any help will be greatly appreciated! Thanks in advance