I'm trying to prove why the mean of the distribution of sums of the top 3 out of 4 fair 6 sided dice is rolls 12.25. Anybody who's rolled a D&D character knows the idea.
$r_n = Rand([1,6])$
$x = \frac{\sum_{i=1}^4{r_i} - min(r_i)}{3}$
Pardon the notation, I wasn't sure how to properly define the problem.
So, I came to derive 12.25 with a computer program that just does several million iterations and comes up with something that's approaching 12.25. I just don't know why or how to prove it. I thought of splitting the interval [1,6] into 4 equal subsets and add the midpoint of the top 3. But that didn't work. Can someone explain why it's 12.25 and how to prove it?