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).

[4] The distribution is named after Carl Friedrich Gauss, who derived it around 1800,[5] and Rodion Kuzmin, who gave a bound on the rate of convergence in 1929.

[6][7] It is given by the probability mass function 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.

In 1928, Kuzmin gave the bound In 1929, Paul Lévy[8] improved it to Later, Eduard Wirsing showed[9] 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.