Question

Benford’s law predicts the frequency distribution of leading digits in many naturally occurring sets of numerical data. The law predicts that the smaller digits will occur disproportionately more often as leading significant digits. This can be used to identify anomalies in the stock market, among other things. The function is as follows:

P(X=x)=P(X= firstdigitofx)=log10(x+1)x

(a) Show this is a legitimate probability distribution.

(b) Now calculate out the probability distribution. How does this compare to if each digit 1-9 had an equal chance of being picked?

(c) Compute the cumulative distribution function.

(d) Using the CDF, what is the probability that the leading digit is at most 3? What about at least 5?

Answer 1

Let X be the random variable denoting the first digit according to Benford's Law.

(a)  Expected value of the first digit when the first digit follows Benford's law = E(X) = 1*0.301+2*0.176+3*0.125+4*0.097+5*0.079+6*0.067+7*0.058+8*0.051+9*0.046 = 3.441

(b) Expected value of the first digit when the possible first digits are equally likely

(c) Variance of the first digit when the first digit follows Benford's Law

Standard deviation of the first digit when the first digit follows Benford's Law

Please provide screenshot of Excel document for Question 2 and 4.

(d)

Under Benford's law, Probability of a leading digit of 7, 8, or 9 = P(X=7) + P(X=8) + P(X=9) = 0.058 + 0.051 + 0.046 = 0.155