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MATH2801/2901 (Higher) Theory of Statistics Semester 1, 2010 Assignment 2 - Transformations, Convergence, Inference Please include this cover sheet...

1. Benford’s Law is named after an American engineer Frank Benford, who in 1938 set
out to ﬁnd the relative frequency of the leading digit in naturally occurring numbers.
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,
= Y
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
c
∈ [1, 10). Use good choices for x and c to give the required result.
MATH2801/2901 (Higher) Theory of Statistics Semester 1, 2010 Assignment 2 – Transformations, Convergence, Infer- ence Please include this cover sheet with your submission. Submission date: Monday lecture, 10am, week 10. Assignment length: No more than 8 pages , including this cover sheet. Q1 /14 Q2 /23 Total /37 Name: Student number: I declare that this assessment item is my own work, except where acknowledged, and has not been submitted for academic credit elsewhere, and acknowledge that the assessor of this item may, for the purpose of assessing this item: Reproduce this assessment item and provide a copy to another member of the University; and/or, Communicate a copy of this assessment item to a plagiarism checking service (which may then retain a copy of the assessment item on its database for the purpose of future plagiarism checking). I certify that I have read and understood the University Rules in respect of Student Academic Misconduct. Signed: Date: 1
1. Benford’s Law is named after an American engineer Frank Benford, who in 1938 set 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 f j , as listed below. j 1 2 3 4 5 6 7 8 9 f j 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 f X . (a) Consider Y = cX , where for convenience, assume that c is any constant with 1 < c < 10. Write down the pdf f Y of Y , in terms of f X . (b) The range of values for Y is [ c, 10 c ), 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 , = Y 10 if Y 10 . The range of values of Z is now [1 , 10), as required. Find an expression for the pdf f Z of Z . (c) If the original pdf f X represents the distribution of naturally occurring numbers on [1 , 10), it must be invariant to scale changes, so that f Z = f X . Use this fact, and the result in part (b), to deduce the form of f X . Hint: the equation f Z ( x ) = f X ( x ) will depend on c , and holds for all x and all c [1 , 10). Use good choices for x and c to give the required result. (d) Hence calculate the relative frequencies in Benford’s Law, ie f j = Z j +1 j f X ( x ) dx, for j = 1 , 2 ,..., 9 . 2
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