Gauss–Kuzmin distribution

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Gauss–Kuzmin
Probability mass function
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Cumulative distribution function
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Parameters (none)
Support k \in \{1,2,\ldots\}
Probability mass function (pmf) -\log_2\left[ 1-\frac{1}{(k+1)^2}\right]
Cumulative distribution function (cdf) 1 - \log_2\left(\frac{k+2}{k+1}\right)
Mean +\infty
Median 2\,
Mode 1\,
Variance +\infty
Skewness (not defined)
Excess kurtosis (not defined)
Entropy 3.4325275...\,[1][2]
Moment-generating function (mgf)
Characteristic function

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_2\left[ 1-\frac{1}{(k+1)^2}\right].

[edit] See also

[edit] References

  1. ^ N. Blachman, "The continued fraction as an information source (Corresp.)", IEEE Transactions onInformation Theory, 30 (1984) pp.671 – 674
  2. ^ Peter Kornerup, David Matula (July 1995). "LCF: A lexicographic binary representation of the rationals". Journal of Universal Computer Science 1: pp. 484–503. 
  • K. Ivan Babenko (1978). "On a problem of Gauss". Soviet Math. Dokl. 19: pp. 136–140. 
  • David H. Bailey, Jonathan M. Borwein, Richard E. Crandall (1995). "On the Khinchine constant". 
  • Carl Friedrich Gauss, "Eine Aufgabe der Wahrscheinlichkeitsrechnung," (1800), In Werke Sammlung, Band 10 Abt 1, pp. 552–556
  • R. O. Kuz'min (1928). "On a problem of Gauss". Atti del Congresso Internazionale dei Matematici, Bologna 6: pp. 83–89. 
  • Paul Lévy (1929). "Sur la loi de probabilité dont dépendent les quotients complets at incomplets d'une fraction continue". Bullitin Societe Mathematique de France 55: pp. 867–870. 
  • Eric W. Weisstein, Gauss–Kuzmin Distribution at MathWorld.
  • Eduard Wirsing (1974). "On the theorem of Gauss–Kusmin–Lévy and a Frobenius-type theorem for function spaces". Acta Arithmetica 24: pp. 507–528. 
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Probability distributions
 
Discrete univariate with infinite support