O'Reilly logo

Classic Problems of Probability by Prakash Gorroochurn

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

Problem 27

Benford and the Peculiar Behavior of the First Significant Digit (1938)

Problem. In many naturally occurring tables of numerical data, it is observed that the leading (i.e., leftmost) digit is not uniformly distributed among {1, 2, . . ., 9}, but the smaller digits appear more frequently that the larger ones. In particular, the digit one leads with a frequency close to 30% whereas the digit nine leads with a frequency close to only 4%. Explain.

Solution. We make use of the fact that, for a given data set, changing the units of measurements should not change the distribution p(x) of the first significant digit. That is, this distribution should be scale invariant:

img

where k and C are constants. Integrating on both sides,

img

Therefore img. Let us now differentiate the latter w.r.t. k:

img

Writing u = kx, we have img so that

img

is a solution. Let D be the first significant digit in base 10. Then ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required