TL;DR: I think @sipa used Beta where he should have used Gamma Binomial.
It’s suspicious that the variance denominator jumps from k-2 to k when I “increase k by 1”.
That is, we have
W = (\text{#S}-1) \times \frac{2^{256}}{H}
and I say “let’s increase #S by 1” by pretending T was a hash. It’s just a little higher than H by definition. I believe this is legitimate because I believe the mean interval from H to T is equal to the mean interval between the elements in #S. So I’ll add T as an extra “hash” to the set S to make S’.
W = (\text{#S}'-1) \times \frac{2^{256}}{T} = \text{#S} \times \frac{2^{256}}{T}
If difficulty is a constant at T, then b = #S’-1 and the W equation for the Beta distribution is equal to the Poisson’s. But the variance denominators are supposedly different, i.e. 1/b verses 1/(#S’ - 2).
So I wonder if we’re back to the original point of this thread, i.e. an (N-1)/N correction is needed when we try to apply a distribution that counts events in a given amount of time in situations where we make the mistake of starting with a given number of events. Or there could be a vice versa mistake. In this case, the Poisson counts events in a fixed amount of time and the Beta distribution counts time for a fixed number of blocks, so there’s a probable mismatch.
In asking Grok about this, it appears the Erlang (or Gamma) should be used in place of the Poisson if we look back a fixed number of blocks, or the Binomial used in place of the Beta if we’re looking back a fixed amount of time. (Poisson is for fixed time and Beta is for fixed blocks). I believe we’re doing the latter, and if I knew how to apply the Binomial I would get “lowest hashes” method to have n^2 / (#S’-1) variance, same as Poisson’s n^2 / b. In other words, the Nth lowest hash is precisely like a difficulty target for N-1 “blocks found” so that we can just use Poisson in both cases. This is a given if my supposition above about the mean interval is correct.
If the difficulty isn’t constant, it still looks correct. In that situation b > #S’ so that the W from b has a lower variance while still using the same Poisson equations for W and Var for both b and #S’.