Gauss–Kuzmin distribution

Probability distribution
name =Gauss–Kuzmin
type =mass
parameters =(none)
support =$k\; in\; \{1,2,ldots\}$
pdf =$-log\_2left\; [\; 1-frac\{1\}\{(k+1)^2\}\; ight]$
cdf =$1\; -\; log\_2left(frac\{k+2\}\{k+1\}\; ight)$
mean =$+infty$
median =$2,$
mode =$1,$
variance =$+infty$
skewness =(not defined)
kurtosis =(not defined)
entropy =$3.4325275...,$ [N. Blachman, "The continued fraction as an information source (Corresp.)", "IEEE Transactions onInformation Theory", 30 (1984) pp.671 – 674] cite journal
author=Peter Kornerup, David Matula
title=LCF: A lexicographic binary representation of the rationals
journal=Journal of Universal Computer Science
month = July
year=1995
volume=1
pages= pp. 484–503]
mgf =
char =
In mathematics, the Gauss–Kuzmin distribution gives the probability distribution of the occurrence of a given integer in the continued fraction expansion of an arbitrary real number. The distribution is named after Carl Friedrich Gauss, who first conjectured and studied the distribution around 1800, and R. O. Kuz'min, who, in 1928, along with Paul Lévy, in 1929, was able to prove Gauss's conjecture. Later, K. Ivan Babenko and Eduard Wirsing completely solved the problem, and were able to show that the speed of convergence of the continued fraction digits to the limiting distribution was exponential.

The probability that any term $A$ in a continued fraction expansion is equal to "k" is given by

:$Pr(A=k)=-log\_2left\; [\; 1-frac\{1\}\{(k+1)^2\}\; ight]\; .$

ee also

* Khinchin's constant
* Gauss–Kuzmin–Wirsing operator

References

:* cite journal
author=K. Ivan Babenko
title=On a problem of Gauss
journal=Soviet Math. Dokl.
year=1978
volume=19
pages= pp. 136–140:* cite journal|author=David H. Bailey, Jonathan M. Borwein, Richard E. Crandall
title= [http://www.reed.edu/~crandall/papers/95-036-Bailey-Borwein-Crandall.pdf On the Khinchine constant]
journal=
year=1995
volume=
pages=:* Carl Friedrich Gauss, "Eine Aufgabe der Wahrscheinlichkeitsrechnung," (1800), In [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN235957348 "Werke Sammlung", Band 10 Abt 1] , [http://dz1.gdz-cms.de/no_cache/en/dms/load/img/?IDDOC=138851 pp. 552–556] :* cite journal
author=R. O. Kuz'min
title=On a problem of Gauss
journal=Atti del Congresso Internazionale dei Matematici, Bologna
year=1928
volume=6
pages= pp. 83–89:* cite journal
author=Paul Lévy
title=Sur la loi de probabilité dont dépendent les quotients complets at incomplets d'une fraction continue
journal=Bullitin Societe Mathematique de France
year=1929
volume=55
pages= pp. 867–870:* MathWorld|title=Gauss–Kuzmin Distribution|urlname=Gauss-KuzminDistribution:* cite journal
author=Eduard Wirsing
title=On the theorem of Gauss–Kusmin–Lévy and a Frobenius-type theorem for function spaces
journal=Acta Arithmetica
year=1974
volume=24
pages= pp. 507–528