Question

Benford's law, also known as the first‑digit law, represents a probability distribution of the leading significant digits of numerical values in a data set. A leading significant digit is the first occurring non‑zero integer in a number. For example, the leading significant digit in the number 127127 is 11. Let this leading significant digit be denoted ?x.

Benford's law notes that the frequencies of ?x in many datasets are approximated by the probability distribution shown in the table.

?x	11	22	33	44	55	66	77	88	99
?(?)P(x)	0.3010.301	0.1760.176	0.1250.125	0.0970.097	0.0790.079	0.0670.067	0.0580.058	0.0510.051	0.0460.046

Determine ?(?)E(X), the expected value of the leading significant digit of a randomly selected data value in a dataset that behaves according to Benford's law? Please give your answer to the nearest three decimal places.

?(?)E(X) =

Select the statement that best describes the interpretation of the expected value of the Benford's law probability distribution.

a.The expected value is sum of all possible leading digits divided by the number of possible leading digits.

b. The expected value is the average value of the leading significant digits in any set of numerical values that follow Benford's law.

c. The expected value is the most common value of the leading significant digit in a set of numerical values.

d. The expected value is sum of the probabilities associated with each leading digit divided by the number of possible leading digits.

e.The expected value is the long‑run average value of the leading significant digits of a set of numerical values.

Answer 1

Solution:

   We are given that: probability distribution of the leading significant digit of a randomly selected data value in a dataset that behaves according to Benford's law.

x	1	2	3	4	5	6	7	8	9
P(x)	0.301	0.176	0.125	0.097	0.079	0.067	0.058	0.051	0.046

We have to find Expected value of x. that is E(X)=........?

To find expected value, we need to find following table:

x	P(x)	x * P(x)
1	0.301	0.301
2	0.176	0.352
3	0.125	0.375
4	0.097	0.388
5	0.079	0.395
6	0.067	0.402
7	0.058	0.406
8	0.051	0.408
9	0.046	0.414

Thus Expected value is:

The statement that best describes the interpretation of the expected value of the Benford's law probability distribution is:

e.The expected value is the long‑run average value of the leading significant digits of a set of numerical values.