Example 4.29

EXAMPLE 4.29

The mean of first digits that follow Benford’s law. Here is the distribution of the first digit for data that follow Benford’s law. We use the letter V for this random variable to distinguish it from the one that we studied in Example 4.28. The distribution of V is

First digit V	1	2	3	4	5	6	7	8	9
Probability	0.301	0.176	0.125	0.097	0.079	0.067	0.058	0.051	0.046

The mean of V is

μ_V = (1)(0.301) + (2)(0.176) + (3)(0.125) + (4)(0.097) + (5)(0.079)

+ (6)(0.067) + (7)(0.058) + (8)(0.051) + (9)(0.046)

= 3.441

The mean reflects the greater probability of smaller first digits under Benford’s law than when first digits 1 to 9 are equally likely.