Here's a problem I've been working on today:

Given a set of finite discrete random variables A_{1}, A_{2}, ..., Athat are mutually independent, what is the distribution of the maximum? In other words, for some_{n}v, what is Pr(max(A_{1}, A_{2}, ..., A)=_{n}v)? Compute the entire distribution in low order time, preferably O(|A_{1}|+ |A_{2}| + |A|)._{n}

Previously, I had the following solution:

Pr(max= v)= Pr("all inputs ≤ v" ∩ "any input =v")= Pr("all inputs ≤ v")•P("any input =v")

That assumes that "all inputs ≤ *v*" and "any input = *v*" are independent. It seemed logical since you don't need to know one to compute the other. Based on this test I made in Excel, I believe I was wrong—or at least my computation above didn't work out.

Irrespective of original problem, I'm trying to understand how you would reason why those two statements are not dependent, and how you would correctly compute the conjunction of the two.