numbers. Benford's law also describes the exponential distribution and the ratio distribution of two exponential distributions well. The fit of chi-squared distribution depends on the degrees of freedom (df) with good agreement with df = 1 and decreasing agreement as the df increases. The F-distribution is fitted well for low degrees of freedom. With increasing dfs the fit decreases but much more slowly than the chi-squared distribution. The fit of the log-normal distribution depends on the mean and the variance of the distribution. The variance has a much greater effect on the fit than does the mean. Larger values of both parameters result in better agreement with the law. The ratio of two log normal distributions is a log normal so this distribution was not examined. Other distributions that have been examined include the Muth distribution, Gompertz distribution, Weibull distribution, gamma distribution, log-logistic distribution and the exponential power distribution all of which show reasonable agreement with the law. The Gumbel distribution – a density increases with increasing value of the random variable – does not show agreement with this law. Generalization to digits beyond the first It is possible to extend the law to digits beyond the first. In particular, for any given number of digits, the probability of encountering a number starting with the string of digits n of that length discarding leading zeros is given by For example, the probability that a number starts with the digits 3, 1, 4 is , as in the figure on the right. Numbers satisfying this include 3.14159..., 314285.7... and 0.00314465... . This result can be used to find the probability that a particular digit occurs at a given position within a number. For instance, the probability that a "2" is encountered as the second digit is And the probability that d (d = 0, 1, ..., 9) is encountered as the n-th (n > 1) digit is The distribution of the n-th digit, as n increases, rapidly approaches a uniform distribution with 10% for each of the ten digits, as shown below. Four digits is often enough to assume a uniform distribution of 10% as "0" appears 10.0176% of the time in the fourth digit, while "9" appears 9.9824% of the time. Moments Average and moments of random variables for the digits 1 to 9 following this law have been calculated: mean 3.440 variance 6.057 skewness 0.796 kurtosis −0.548 For the