GENERAL ¨ ARTICLE
The First Digit 1
Tanya Kaushal Srivastava
Tanya Kaushal Srivastava
is a BS-MS student at the
Department of Mathematics, IISER Mohali and is a
KVPY fellow.
Keywords
Random numbers, unpredictable, equidistribution, W eyl criterion, absorptive property.
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Table 1. First digits in the
newspaper Times of Chandigarh.
First
Digit
1
2
3
4
5
6
7
8
9
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Frequency
40%
20%
7%
7%
5%
3%
9%
2%
5%
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GENERAL ¨ ARTICLE
Table 2. The first digits distribution according to Benford’s
law.
First
Digit
1
2
3
4
5
6
7
8
9
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Frequency
30.1 %
17.6 %
12.4 %
9.7 %
7.9 %
6.7 %
5.8 %
5.1 %
4.5 %
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If X obeys
Benford’s law, and
Y is any positive
random variable
independent of X,
then the product
(XY) also obeys
Benford’s law,
without any further
assumptions on Y.
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Figure 1 (left). Cumulative
distribution function of uniform distribution on [0,1].
Figure 2 (right). Probability
density function of uniform
distribution [0,1].
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Figure 3. Cumulative distribution function of {log Y}.
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Figure 3. Cumulative distribution function of {
1082
}.
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This law also has
many applications,
especially in
experimental sciences
where the underlying
mechanism has some
exponential growth,
which is usually the
case in many
biological
experiments.
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GENERAL ¨ ARTICLE
Suggested Reading
[1] Aditi Kar, Weyl's Equidistribution Theorem, Resonance, Vol.8, No.5,
pp.30–37, 2003.
[2] S Kunoff, N! has the First Digit Property, The Fibonacci Quarterly,
Vol.25, No.4, pp.365–-367, 1987.
[3] R A Raimi, The First Digit Problem, Scientific American, Vol.221, No.6,
pp.109–120, 1969.
[4] Sheldon Ross, A First Course in Probability, Pearson Education, Sixth
Edition, 2009.
[5] T Tao, Benfords law, Zipfs law, and the Pareto distribution, http://
terrytao.wordpress.com/2009/07/03/benfords-law-zipfs-law-and-thepareto-distribution/, 2009.
[6] L C Washington, Benford's Law For Fibonacci And Lucas Numbers,
Address for Correspondence
Tanya Kaushal Srivastava
Department of Mathematics
Indian Institute of Science
Education and Research
Mohali (IISER)
Mohali 140 306, India
Email:
[email protected]
The Fibonacci Quarterly, Vol.19, pp.175–177, 1981.
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