Classroom Companion: Business

Definition 16.2 Heavy Tail Distribution and Power-Law Distribution
The discrete heavy tail distribution is a statistical distribution in which the tail of the 
distribution falls off more slowly than exponential decay. Any distribution of the 
form f(X = k)~k
−γ
, where γ > 1 is a constant and the positive integer k is the running 
variable, is a discrete heavy tail distribution. This particular distribution is also called 
a discrete power-law distribution (Schroeder, 
2009
). The symbol f(X = k) stands for 
“the frequency of occurrences that X is equal to k,” and the tilde (~) means “is pro-
portional to.” If γ is small (between 1 and 2), then it is also called a long tail 
 distribution.
Anderson, based on the works by Brynjolfsson et al., found that the popularity of 
books sold by Amazon seemed to follow a rank distribution in accordance with 
Zipf ’s law; that is, a long tail distribution with γ = 1 and where the probability 
drops to zero for all k larger than a certain upper threshold (the cutoff). In 2000, 
the number of titles available on Amazon was 2.3 million, while ordinary book-
16.3 · Numerical Analysis of the Long Tail

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stores held, on average, about 40,000 titles. The “long tail” of Amazon then con-
sisted of more than 2.2 million books, that is, books available on Amazon, but not 
in an ordinary bookstore.
Let us then apply Zipf ’s law to the sale of books. The number of titles held by 
Amazon is 2.3 million books, while an ordinary bookstore holds about 40,000 
titles. The titles held by an ordinary bookstore are, according to Zipf ’s law, the 
most popular titles, ranging from 1 to 40,000 in popularity. The titles with popular-
ity ranging from 40,001 to 2,300,000 in popularity are the long tail. The relative 
number of sales of books from the long tail is then:
k
k
k
k
k
k
k
k









 
1
2 300 000
1
40 000
1
2 300 000
1
40 000
1
1
1
1
1
,
,
,
,
,
,
k
k
k


 



1
2 300 000
1
1
40 000
2 300 000
26 6
,
,
ln
,
ln ,
,
. %.