out to ﬁnd the relative frequency of the leading digit in naturally occurring numbers.
He was curious about this upon observing that the ﬁrst pages of books of logarithm
tables were much more worn and dirty than later pages – this was at a time when
complex multiplications in engineering practice were carried out using such tables, a
long time before the era of electronic calculators and computers. His ﬁndings were
based upon observing about 20,000 numbers.
Benford’s Law states that the leading digit j of naturally occurring numbers occur
with relative frequency fj , as listed below.
j 1 2 3 4 5 6 7 8 9
fj 0.301 0.176 0.125 0.0969 0.0792 0.0669 0.0580 0.0512 0.0458
The task in this question is to carry out a theoretical derivation of Benford’s Law.
The method for doing this is based on the idea of invariance to changes of scale, or
equivalently, to changes of units for measured quantities.
Consider a random number X , which is positioned within the interval [1, 10) by shifting
the decimal point, if necessary. Its leading digit is j if j
≤ X < j + 1. Let the pdf of
X be fX .
(a) Consider Y = cX , where for convenience, assume that c is any constant with
1 < c < 10. Write down the pdf fY of Y , in terms of fX .
(b) The range of values for Y is [c, 10c), so to move this back to [1, 10), values of
Y > 10 must have the decimal point moved one place to the left. Thus, deﬁne
Z = Y if Y < 10,
10 if Y ≥ 10.
The range of values of Z is now [1, 10), as required. Find an expression for the
pdf fZ of Z .
(c) If the original pdf fX represents the distribution of naturally occurring numbers
on [1, 10), it must be invariant to scale changes, so that fZ = fX . Use this fact,
and the result in part (b), to deduce the form of fX .
Hint: the equation fZ (x) = fX (x) will depend on c, and holds for all x and all
∈ [1, 10). Use good choices for x and c to give the required result.
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