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. Call the first digit of a randomly chosen record $X$for short. Benford’s law gives this probability model for$X$(note that a first digit can’t be $0$)

(a) Show that this is a legitimate probability distribution.

(b) Make a histogram of the probability distribution. Describe what you see.

(c) Describe the event $X\ge 6$in words. What is $P(X\ge 6)$?

(d) Express the event “first digit is at most $5$” in terms of $X$. What is the probability of this event?

Short answer

a) The probability range from $0$to $1$.

b)

c) As a result, $0.222$is the required probability.

d) As a result, $0.778$is the required probability.

Step-by-step solution

Part(a) Step 1: Given Information

Given that,

Part(a) Step 2: Explanation

Calculate the probability sum as follows:

Sum of probabilities

$$\begin{gathered} =0.301+0.176+\dots .+0.046 \\ =1 \end{gathered}$$

The probability range from $0$ to $1$, and the sum of the probabilities equals $1$. As a result, the probability distribution supplied is a valid probability distribution.

Part(b) Step 1: Given Information

Given that,

Part(b) Step 2: Explanation

The graph might be made as follows:

The graph's right side contains the majority of the data. As a result, the distribution is slanted right.

Part(c) Step 1: Given Information

Given that,

To explain: The $X\ge 6$. Calculate the $P(X\ge 6)$ as well.

Part(c) Step 2: Explanation

$6$or more digits are implied by $X\ge 6$. It simply signifies that there are at least six digits.

$P(X\ge 6)$may be determined using the following formula:

$$\begin{gathered} \begin{array}{c}P(X\ge 6)=P(X=6)+P(X=7)+P(X=8)+P(X=9)\end{array} \\ =0.067+0.058+0.051+0.046 \\ =0.222 \end{gathered}$$

Part(d) Step 2: Given Information

Given that,

Part(d) Step 2: Explanation

In terms of $X$, the event "first digit is at most $5$" may be expressed as $X\le 5$. $P(X\le 5)$can be computed as:

$\begin{array}{l}P(X\ge 5)=P(X=1)+P(X=2)+.........+P(X=5)\\ =0.301+0.176+........+0.079\\ =0.778\end{array}$