Each ball is a sum of many small random steps. The Central Limit Theorem says such sums become Gaussian once you add enough of them โ nearly whatever the steps look like.
A ball falls through n peg rows, going left/right with
probability ยฝ. Its slot counts the rights: k ~ Binomial(n, ยฝ),
mean n/2, variance n/4 โ a bell as n grows.
Pegs only add ยฑ1 steps, so this mode drops them and sums n draws
from a source you pick. Each draw uses inverse-transform sampling:
U ~ Uniform(0,1), then X = Fโปยน(U) (shown in the inset).
The standardized sum z = (S โ nฮผ)/(ฯโn) still lands on the same bell.
The CDF view shades the gap between the empirical and
normal CDFs. D is the largest gap; the p-value comes from the
Kolmogorov distribution at โNยทD. We standardize with the
known ฮผ, ฯ, so the test is valid. (The quincunx is discrete, so for
small n, D settles above zero โ use more rows.)
Space pause ยท R reset ยท V board โ CDF ยท F full-screen ยท H help.