Or rather, under­e­d­u­cated semi-​​pro.

I’ve got a com­bi­na­to­r­ial opti­miza­tion prob­lem. It’s deter­min­is­tic, and rather arbi­trary, so don’t worry about the details.

There are a lot of fea­si­ble solu­tions. I drew a sam­ple of 100000 solu­tions from the 82318282158320505 fea­si­ble ones, uni­formly, i.i.d., all that stuff. “Ran­dom sam­ples” say, and yes, I know that’s a loaded phrase.

The fitness/​cost/​performance of these 100000 sam­ples appears to fol­low a nor­mal dis­tri­b­u­tion very nicely indeed. Yes, I know that’s a loaded phrase.

Say the aver­age is 90000, and the mea­sured stan­dard devi­a­tion is 1000.

I know there are 82318282158320505 solutions.

Assum­ing the right stuff for this ques­tion to make sense: What’s the expected value of the lowest-​​cost (first) of the 82318282158320505 solutions?

I can look at MathWorld’s expla­na­tion of the Extreme Value Distribution(s) and the wikipedia bit on Order Sta­tis­tics until I’m blue in the face and my head’s all tilty and like, “Wha?” but in the end, I am just a dumb ol’ biol­o­gist and can’t make the num­bers and the equa­tions click together.

I see that the for­mu­lae include places to stick whatcha call yer mean, and yer stad dev, and stuff. And an N, which is the num­ber of sam­ples you draw from said nor­mal dis­tri­b­u­tion. My N, she is too beeg.

Any­body?

Bueller?