# Cumulative Probability Distribution of Maximum and 2nd from Maximum of 4 Variables

I understand that the cumulative probability distribution cum(x) of the maximum of 2 variables x1 and x2 with probability distribution p1(x1) and p2(x2) is the product of the two cumulative probability distributions.

So if the cumulative probability distribution of p1(x1) is cum1(x1), and the cumulative probability distribution of p2(x2) is cum2(x2), the cumulative distribution of the maximum of x1 and x2 is cum_max(x) = cum1(x) * cum2(x).

In my case I have 4 independent variables, with identical probability distribution p(x) and cumulative probability distribution cum(x). I want to determine the cumulative distribution function of the maximum of all 4, cum_max. and I also want to determine the cumulative distribution function of the 2nd from maximum, cum_2nd.

By iterating the property of the product of cumulative probability distribution I obtain: cum_max(x) = [cum(x)]^4

I am stomped about what relationship applies to the 2nd from maximum.

After posting, I found a reference in this forum pointing to Wolfram's Order statistics. The solution supposedly is $$4*cum^3-3*cum^4$$