Sheppa28

Short page for now, might expand when given more time:
MS in Statistics; homepage here

Benford distribution

Although it was historically discovered to be an empirical statement, it is mathematically a true probability distribution, and as such, characteristics of that distribution can be determined for theoretical reasons

Characteristic	Formula (general)	b=10
Probability mass function	$\log _{b}\left(1+{\frac {1}{d}}\right)$	$\log _{10}\left(1+{\frac {1}{d}}\right)$
Cumulative distribution function	$\log _{b}\left(1+\lfloor d\rfloor \right)$	$\log _{10}\left(1+\lfloor d\rfloor \right)$
Expected value	$b-log_{b}\left(b!\right)$	~ 3.44024
Median	$\lceil {\sqrt {b}}\rceil -1$	3
Mode	1	1
Variance	$\left({b^{2}-(2b-1){log_{b}\left(b!\right)}+2\sum _{k=0}^{b-1}{log_{b}\left(k!\right)}}\right)-\mu ^{2}$	~ 6.05651

For an arbitrary base b, the probability that d (d = 0, 1, ..., b-1) is encountered as the n-th (n>1) digit can be expressed in closed-form expression as

$\log _{b}\left({\frac {\Gamma \left({\tfrac {1+b^{n}+d}{b}}\right)\Gamma \left({\tfrac {b^{n}+bd}{b^{2}}}\right)}{\Gamma \left({\tfrac {b+b^{n}+bd}{b^{2}}}\right)\Gamma \left({\tfrac {b^{n}+d}{b}}\right)}}\right)$

Where $\Gamma (x)$ is the Gamma function