Exercise 19 investigated Benford’s law, a discrete distribution with pmf given by p(x) = log10((x + 1)/x) for x = 1, 2, …, 9. Use the inverse cdf method to write a program that simulates the Benford’s law distribution. Then use your program to estimate the expected value and variance of this distribution.

Data From Exercises 19

Suppose that you read through this year’s  issues of the New York Times and record each  number that appears in a news article—the  income of a CEO, the number of cases of  wine produced by a winery, the total charitable  contribution of a politician during the  previous tax year, the age of a celebrity,  and so on. Now focus on the leading digit  of each number, which could be 1, 2, …, 8,  or 9. Your first thought might be that the  leading digit X of a randomly selected  number would be equally likely to be one  of the nine possibilities (a discrete uniform  distribution). However, much empirical  evidence as well as some theoretical arguments  suggest an alternative probability distribution called Benford’s law:

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(x + ¹) X p(x) = P(1st digit is x) = log10 x = 1,2, ..., 9