Question

Benford’s law Faked numbers in tax returns, invoices, or expense account claims often display patterns that aren’t present in legitimate records. Some patterns, like too many round numbers, are obvious and easily avoided by a clever crook. Others are more subtle. It is a striking fact that the first digits of numbers in legitimate records often follow a model known as Benford’s law. 4 Call the first digit of a randomly chosen legitimate record X for short. The probability distribution for X is shown here (note that a first digit cannot be 0).

Part (a.) A histogram of the probability distribution is shown. Describe its shape.

Part (b). Calculate and interpret the expected value of X.

Short answer

Part (a)

Skewed to the right

Unimodal

1 is the most common first digit

The first digits range from 1 to 9

Part (b)$\mu =3.441$

Step-by-step solution

Part (a) Step 1. Given information

The probability distribution for X is shown here

Part (a) Step 2. Describe its form.

Because the highest bar in the histogram is to the left, and there is a tail of smaller bars to its right, the distribution is skewed to the right.

Because there is only one peak in the histogram, the distribution is unimodal.

Because the highest bar in the histogram is centered at 1, the most common first digit is 1.

The first digits are numbered from 1 to 9.

As a result:

Skewed to the right

Unimodal

1 is the most common first digit

The first digits range from 1 to 9

Part (b) Step 1. Interpret the expected value of X. 

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

The expected value (or mean) is the sum of the product of each possibility 2, with its probability $p_i$

$$\begin{gathered} \mu =\sum _{}{x}_i{p}_i \\ \mu =1x0.301+2x0.176+3x0.125+4x0.097+5x0.079 \\ +6x0.067+7x0.058+8x0.051+9x0.046 \\ \mu =3.441 \end{gathered}$$

The first digit of a randomly chosen legitimate record is 3.441 on average.