Benford’s law , which concerns the spread of leading significant digits, states that the first (non-zero) digit in a wide range of number collections isn’t uniform for the number 1 through 9. Instead, the numbers follow the non-uniform **Benford distribution**.

## Use of the Benford Distribution

While popular for analyzing numbers in texts, the law also appears in a wide range of probability and statistics applications, including in products of i.i.d. random variable mixtures of random samples and in some stochastic models [1]. In real life, the distribution can model most accounting data, census statistics, and stock market data [2]; A specific use of the distribution is to audit financial records. The numbers in these records should theoretically follow the Benford distribution; if they do not, it is a sign that the records may have been falsified. Benford’s law isn’t widely known outside of statistical circles, so it’s unlikely that anyone falsifying records would know to distribute the fake numbers according to the Benford distribution [3].

The Benford distribution also holds for some parametric survival distributions, likely because many popular parametric lifetime models also follow the distribution for certain parameter values [4].

Data that obeys Benford’s law follows a Benford distribution; If a random variable has the Benford distribution, it can be denoted as **X ∼ Benford** [5].

## Benford Distribution PDF & CDF

The Benford distribution has the probability density function (PDF):

And the cumulative distribution function (CDF):

## References

[1] Berger, A. & Hill, T. A basic theory of Benford’s Law. Probability Surveys Vol. 8 (2011) 1-126.

[2] Hill, T. (1995). A Statistical Derivation of the Significant Digit Law. Statistical Science. 10(4):354-363.

[3] Tam Cho, W. & Gaines, B. Breaking the (Benford) Law: Statistical Fraud Detection in Campaign Finance.

[4] Leemis, L. et al. Survival Distributions Satisfying Benford’s Law. Retrieved November 9, 2021 from: http://www.math.wm.edu/~leemis/2000amstat.pdf

[5] Benford Distribution. Retrieved November 9, 2021 from: http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Benford.pdf