Gauss-Kuzmin distribution: probability of each partial quotient
P(k) = log₂(1 + 1/k(k+2)). The partial quotient 1 appears in ~41% of all continued fraction expansions of random real numbers.
Cumulative probability: fraction covered by digits 1 through k
By k=3 over two-thirds of all partial quotients are accounted for. The sequence converges slowly toward 1.
Continued fraction: nested structure unpacked
x = a₀ + 1/(a₁ + 1/(a₂ + 1/(a₃ + …)))
= a₀ + 1/a₁ + 1/a₁a₂ + … (truncated approximations)
For almost all real x, the geometric mean of a₀, a₁, a₂, … converges to Khinchin's constant K₀ ≈ 2.6854.
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