Gauss–Kuzmin distribution
In mathematics, the Gauss–Kuzmin distribution is a discrete probability distribution that arises as the limit probability distribution of the coefficients in the continued fraction expansion of a random variable uniformly distributed in (0, 1). The distribution is named after Carl Friedrich Gauss, who derived it around 1800, and Rodion Kuzmin, who gave a bound on the rate of convergence in 1929. It is given by the probability mass function Gauss–Kuzmin theorem. Let be the continued fraction expansion of a random number "x" uniformly distributed in (0, 1). Then Equivalently, let then tends to zero as  n tends to infinity. Rate of convergence. In 1928, Kuzmin gave the bound In 1929, Paul Lévy improved it to Later, Eduard Wirsing showed that, for "λ"=0.30366... (the Gauss-Kuzmin-Wirsing constant), the limit exists for every "s" in [0, 1], and the function "Ψ"("s") is analytic and satisfies "Ψ"(0)="Ψ"(1)=0. Further bounds were proved by K.I.Babenko.