I am extremely enthused at finding this website. Thanks to Dilip Sarwate for some comments. I still don't have a final solution, however. Below is edited to better focus the problem.
Here is my problem. The situation is N random variables X(i), and one more, Y. The N+1 variables are independently distributed, with the same type of distribution, but they are not IID; each variable has a different mean. (For example, all N+1 variables might be exponentially distributed). These distribution means are drawn from a known distribution with a small standard deviation. Ultimately, I want to be able to take observed probabilities (e.g., I observe that Y exceeds the max of the X(i), for 1.1% of the 20,000 draws), and infer the mean of Y.
This is for an economics application, ultimately to infer "quality" from market share data. I am seeking functional forms such that these questions can be answered.
Since they are independent, then as Dilip pointed out, it is not difficult to figure out P(X(i)<x) or P(Y<x) but I am having difficulty figuring out P(Y>Max(X(i)). Apologies if this is basic. If substantive theory is involved, I am not averse to including someone as a coauthor.