Full text: Pareto Distribution

In the case of the distribution of towns by size of 
population the rank-size relation has been used ( Zipf 1949 ) 
which is the same as the Pareto distribution except that 
it uses rank as a measure of the tail ( instead of the 
number of twfons above a certain size ) so that the higher 
the rank (beginning with rank one for the largest town ) 
the smaller the size of the town. Zipf believed ( incorrectly) 
that the coefficient is always about one so that 
the product of rank and size is constant. But Pareto, 
of course, was even more "out" wwith his belief that 
the Pareto coefficient for income cx always equals unity. 
In highly industrialised countries to-day it is above 2 
and sometimes above 3. 
The main interest of the Pareto distribution lies not in 
its rather limited use as a measure of inequality but 
inthe explanations it has provoked, naturally so since 
regular patterns are felt to be a challenge to the mind. 
There are two types of approach to the problem, that of 
Champernowne, Yule and Simon which explains the characteristic 
pattern as the steady state of a stochastic process 
which has been evolving in time, so that the pattern reflects 
something which has been going on in the past. In contrast 
to that Mandelbrot has been looking for a "synchronic" 
explanation which does not depend on a process in time. 
He is mainly concerned with the reproductive quality of 
the Pareto distribution: If a large number of independent 
random variables are identically distributed according to 
Paretos law then the sum of these random variables will 
also be distributed according to this law. ^

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