Gauss-Kuzmin Distribution
Continued fraction expansion: x = a₀ + 1/(a₁ + 1/(a₂ + …)). Almost every x has same coefficient distribution.
Sample x:
Random reals
π - 3
e - 2
√2 - 1
φ - 1 (golden)
Samples:
100
1000
5000
Depth:
20
50
100
Compute
Live
Samples:
0
Khinchin K:
2.6854…
Empirical K:
-
P(a=1) theo:
0.4150
P(a=1) emp:
-
Gauss-Kuzmin: P(aₙ=k) = -log₂(1-1/(k+1)²)
Khinchin's constant: ∏(1+1/(k(k+2)))^(log₂ k)
≈ 2.6854520010327…
Almost all x have same K