Gauss–Kuzmin distribution

Gauss–Kuzmin
Probability mass function
Cumulative distribution function
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)
Excess 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

${\displaystyle p(k)=-\log _{2}\left(1-{\frac {1}{(k+1)^{2}}}\right)~.}$