User:BeyondNormality/Benford distribution

Also known as the Bradford-type law, distribution of first digits, distribution of most significant digits, first-digit problem, Law of anomalous numbers, log law of numbers, the probability mass function of the Benford distribution is given by



\mathrm{Benford}(x) \equiv \Pr(X = x) = \frac{\log \left(\frac{1}{x}+1\right)}{\log (10)} $$


 * $$ x = 1, 2, \dots ,n $$

Cumulative Distribution Function
 * $$ F_X(x) = \frac{\log (x+1)}{\log (10)} $$

Expected Value
 * $$ \mathbb E(X) = 10-\frac{\log (3628800)}{\log (10)} = 3.44024 $$

Variance

\operatorname{Var}(X) = 81-\frac{73 \log (2)+50 \log (3)+9 \log (5)+13 \log (7)}{\log (10)}-\left(10-\frac{\log (3628800)}{\log (10)}\right)^2 = 6.05651 $$

Probability Generating Function
 * $$ G(t) = \sum _{x=1}^9 \frac{t^x \log \left(1+\frac{1}{x}\right)}{\log (10)} $$