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

## TL;DR

This paper constructs a smooth probability distribution on positive real numbers that exactly follows Benford's law for leading digits, using trapezoidal rule error theory.

## Contribution

It demonstrates the existence of a smooth distribution on (0,∞) that precisely matches Benford's law, a novel theoretical result.

## Key findings

- Existence of a smooth distribution matching Benford's law exactly
- Use of trapezoidal rule error theory for distribution construction
- Provides a theoretical foundation for Benford's law in continuous distributions

## Abstract

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.