Preprint arXiv:1905.02031 [math.PR]; last accessed June 6, 2019.

**ISSN/ISBN:** Not available at this time.
**DOI:** Not available at this time.

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**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.

**Bibtex:**

```
@ARTICLE{,
author = {{Ozawa}, Kazufumi},
title = "{Continuous Distributions on $(0,\,\infty)$ Giving Benford's Law Exactly}",
journal = {arXiv e-prints},
year = "2019",
month = "May",
eid = {arXiv:1905.02031},
pages = {arXiv:1905.02031},
}
```

**Reference Type:** Preprint

**Subject Area(s):** Probability Theory, Statistics