Gauss–Kuzmin distribution

Gauss–Kuzmin

Parameters	(none)
Support	${\displaystyle k\in \{1,2,\ldots \}}$
pmf	${\displaystyle -\log _{2}\left[1-{\frac {1}{(k+1)^{2}}}\right]}$
CDF	${\displaystyle 1-\log _{2}\left({\frac {k+2}{k+1}}\right)}$
Mean	${\displaystyle +\infty }$
Median	${\displaystyle 2\,}$
Mode	${\displaystyle 1\,}$
Variance	${\displaystyle +\infty }$
Skewness	(not defined)
Ex. kurtosis	(not defined)
Entropy	3.432527514776...

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

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 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.

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