User: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

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


 * $$\log_b\left({\frac{\Gamma\left(\tfrac{1 + b^n + d}{b}\right)\Gamma\left(\tfrac{b^n + b d}{b^2}\right)}{\Gamma\left(\tfrac{b + b^n + b d}{b^2}\right)\Gamma\left(\tfrac{b^n + d}{b}\right)}}\right)$$

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