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Benford distribution

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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    
Cumulative distribution function    
Expected value   ~ 3.44024
Median   3
Mode 1 1
Variance   ~ 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

 

Where   is the Gamma function