(Considering that this is a fairly narrow-appeal post, I will pipe it over to the right-hand “Meta-Analysis” column shortly.)
Dear PEC readers, I have a math puzzle. It relates to my gerrymandering project. If you are good at working with probability distributions, take a look. Can you solve it?
Here is the puzzle. It is basically a closed-form calculation of the numerical simulations I did for that NYT piece. It is for a peer-reviewed paper I am writing on how to establish criteria for fair Congressional districting.
Partitioning of voters in a state with randomly selected districts. Imagine a state with N districts, and a two-party winner-take-all system (i.e. the U.S. system for electing House members). Select districts at random from a distribution whose vote share v (for party #1) follows a near-Gaussian distribution whose average is A and standard deviation is S.
Now add the condition that the statewide two-party vote yields a fraction F1 of votes for political party #1 (and of course the other party gets F2=1-F1). Therefore districts v1..N must satisfy the constraint sum(v1..N)/N=F1.
What is the probability distribution of k, where k is defined as the number of districts in which vi>0.5? Give the mean and SD of the expected number of seats to be won by party #1. Also describe the degree to which the distribution resembles a Gaussian.
P.S. When F1 is close to A, I believe the answer is approximately <k> = N*p and std(k) = sqrt [N*p*(1-p)], where p = normcdf(F1,0.5,S). If you can do better, let me know!
P.P.S. Here is a rephrasing of the problem: Consider a normally distributed variable with mean mu and standard deviation sigma. Draw from it k times. You only accept sets of draws whose average is constrained to be mu’, which is unequal to mu. What is the distribution of the draws?
P.P.P.S. Probably solved. It’s as above, except instead of normcdf(F1,0.5,S) we have normcdf(F1,0.5,S*sqrt((k-1)/k)). This arises in a semi-obvious way from the derivation of the standard error of the mean.
The gift I have in mind is kind of small: a signed copy of either (or both) of my books. I will see if I can think of something nicer to send…