Benford’s law and fraud A not-so-clever employee decided to fake his monthly expense report. He believed that the first digits of his expense amounts should be equally likely to be any of the numbers from $1$to $9$. In that case, the first digit $Y$of a randomly selected expense amount would have the probability distribution shown in the histogram.

(a). Explain why the mean of the random variable Y is located at the solid red line in the figure.

(b) The first digits of randomly selected expense amounts actually follow Benford’s law (Exercise 5). What’s the expected value of the first digit? Explain how this information could be used to detect a fake expense report.

(c) What’s $P(Y>6)$? According to Benford’s law, what proportion of first digits in the employee’s expense amounts should be greater than $6$? How could this information be used to detect a fake expense report?

Given in the question that, the first digits of his expense amounts should be equally likely to be any of the numbers from $1$to $9$.

The supplied distribution is symmetric (the graph to the left of $5$is a mirror image of the graph to the right of $5$) and the mean is located in the middle (which is $5$).

Given in the question is a graph

Table $5$shows the probability distribution of Benford's law.

The expected value is calculated by multiplying each possibility by its probability:

$$\begin{gathered} E\left(X\right)=\sum xP\left(x\right) \\ =1\times 0.301+2\times 0.176+3\times 0.125+4\times 0.097+5\times 0.079+6\times 0.067+7\times 0.058+8\times 0.051+9\times 0.046 \\ =3.441 \end{gathered}$$

Given that the first digits of his spending amounts might be any number from $1$to $9$,

Using the given uniform distribution, add the associated probabilities:

$$\begin{gathered} P(Y>6)=P(Y=7)+P(Y=8)+P(Y=9) \\ =\frac{1}{9}+\frac{1}{9}+\frac{1}{9}=\frac{3}{9}=\frac{1}{3} \\ \approx 0.3333 \end{gathered}$$

Using Benford's law, add the corresponding probabilities:

$$\begin{gathered} P(Y>6)=P(Y=7)+P(Y=8)+P(Y=9) \\ =0.058+0.051+0.046 \\ =0.155 \end{gathered}$$

Determine the percentage of cost reports with first digits greater than $6$. If this proportion is closer to $0.3333$than to $0.155$, the cost report appears to be false.