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Register a new userFirst digit law from Laplace transform
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Added by
Bo-Qiang Ma
Publication date
2019
fields
Mathematical Statistics
and research's language is
English
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The occurrence of digits 1 through 9 as the leftmost nonzero digit of numbers from real-world sources is distributed unevenly according to an empirical law, known as Benfords law or the first digit law. It remains obscure why a variety of data sets generated from quite different dynamics obey this particular law. We perform a study of Benfords law from the application of the Laplace transform, and find that the logarithmic Laplace spectrum of the digital indicator function can be approximately taken as a constant. This particular constant, being exactly the Benford term, explains the prevalence of Benfords law. The slight variation from the Benford term leads to deviations from Benfords law for distributions which oscillate violently in the inverse Laplace space. We prove that the whole family of completely monotonic distributions can satisfy Benfords law within a small bound. Our study suggests that Benfords law originates from the way that we write numbers, thus should be taken as a basic mathematical knowledge.
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