I have $ B $ bins and $ G $ balls, and each of the balls has a weight. I toss the balls uniformly into the bins. I select the $ K $ bins that have the highest weights as determined by the weights of the balls that are in them, and I then record the total number of balls that appear in these $ K $ bins.
I want to model the sum of the number of balls that appear in these $ K $ bins. Can I create a "worst-case" model of this quantity? E.g., can I say that this quantity's variance can be no worse than some other computable value?
Intuitively, I believe that I can because, as $ K \rightarrow B $, this quantity's expected value becomes $ G $ and its variance goes to 0.