91影视

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:

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

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:

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

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.