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So the question remains, how can we justify all of these empirical observations of the law in a more rigorous mathematical way?
First of all, an important point to note is that when we sample values from some common probability distributions (like the exponential distribution and the log normal distribution) the leading digits that you get already come close to satisfying Benford’s Law (see the graphs at the bottom of the article).
But since this operation is not allowed to change the probability of leading digits, that means that the probability of having a leading digit of 1 must be the same as the probability of having a leading digit of any of 5, 6, 7, 8 or 9.
This property is satisfied by the formula given above, since Of course, there is nothing special about multiplying the numbers from our random source by 2, so a similar property must hold regardless of what we multiply our numbers by.
For each, the pink line shows what we would expect to get if Benson’s Law held perfectly.As you can see, for some distributions we get a good fit (e.g.the exponential and log normal distributions) whereas for others the fit is poorer (e.g. What the third graph in each table shows is the distribution of leading digits that we get when, instead of sampling just from one copy of each distribution, we sample from 9 different copies (with equal probability), each of which has a different variance (in most cases chosen to be proportional to the values 1 up to 9).Note that if we were writing our numbers in a base other than 10 (i.e.