Continuous Distributions on $(0,\mathrm{\infty })$ Giving Benford's Law Exactly

Benford’s law is a famous law in statistics which states that the leading digits of random variables in diverse data sets appear not uniformly from 1 to 9; the probability that d (d=1,…,9) appears as a leading digit is given by \log_{10}(1+1/d). This paper shows the existence of a random variable with a smooth probability density on (0,\infty) whose leading digit distribution follows Benford’s law exactly. To construct such a distribution the error theory of the trapezoidal rule is used.