MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
I assume that you mean joint normal, in which case x_1-x_2 is normal. The second qu. asks if the set of joint normal rv's x_a-x_i are all positive, which is more difficult (requires some numerical integration, I think). – George Lowther May 10 '10 at 21:53
Thanks, I know how to do it via numerical integration, but was hoping there's something simple I'm missing! – user6012 May 10 '10 at 22:05
up vote 2 down vote accepted

Yes. For normal random variables, the probability P(X > Y) can be calculated in closed form. See this post on random inequalities.

Regarding your more general question, see this article on random inequalities with three or more random variables.

share|cite|improve this answer
Thanks, I don't have enough rep to upvote you, but I will look into this. – user6012 May 10 '10 at 21:53

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.