Given two normally distributed variables `x_1, x_2`, is there a non-simulation method of calculating the probability that `x_1 > x_2`?


Generalizing a bit, what is the probability that given a list of normally distributed variables `x_i`, the probability that `x_a = max x_i`?



Comparing normally distributed variables
Yes. For normal random variables, the probability P(X > Y) can be calculated in closed form. See this post on [random inequalities](http://www.johndcook.com/blog/2008/07/26/random-inequalities-ii-analytical-results/).


Regarding your more general question, see this article on random inequalities with [three or more random variables](http://www.johndcook.com/blog/2008/09/06/random-inequalities-vii-three-or-more-variables/).


