Which distribution is likely to be the source of the largest value? Suppose I sample many times independently from each of two normal distributions. Given the means, standard deviations, and number of samples from each distribution, how can I calculate the probability that the highest sampled value came from the first distribution?
I can solve this question by simulating it numerically, but haven't figured out an analytic expression.
 A: The answer works in the same way as jbowman's (now deleted answer) except that X$_1$ and X$_2$ are replaced by M$_1$ and M$_2$ where M$_1$ has the distribution of the maximum of the n$_1$ samples from group 1 and M$_2$ is the maximum of the n$_2$ samples from group 2.  So now we have to replace the distribution of the difference of two independent normals to the distribution of the difference between the maxima for the two normals.  Unfortunately the exact distribution for the maximum is not simple but the answer is to calculate the probability that M$_1$-M$_2$>0. Actually for M$_1$ it is F$_1$$^n$(x) and F$_2$$^n$(x) for M$_2$ where F$_1$ and F$_2$ are the respective normal distributions for group 1 and group 2.  This is explained below in my comment. 
As a further note asymptotically M$_1$ and M$_2$ can be appropriately normalized to have their distributions of the form of a Gumbel distribution (i.e. F(x) =exp(-e$^-$$^x$)). For large n$_1$ and n$_2$ that could be useful in getting an approximate answer to the desired probability.
As requested by the OP here are some links to books on extreme value theory at the amazon.com site.
Gumbel
Leadbetter et al
Galambos
Coles
deHaan
