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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?

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  • $\begingroup$ 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). $\endgroup$
    – George Lowther
    Commented May 10, 2010 at 21:53
  • $\begingroup$ Thanks, I know how to do it via numerical integration, but was hoping there's something simple I'm missing! $\endgroup$
    – user6012
    Commented May 10, 2010 at 22:05

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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.

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  • $\begingroup$ Thanks, I don't have enough rep to upvote you, but I will look into this. $\endgroup$
    – user6012
    Commented May 10, 2010 at 21:53

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