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Chaos and FractalsAn Elementary Introduction$
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David P. Feldman

Print publication date: 2012

Print ISBN-13: 9780199566433

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199566433.001.0001

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When do Averages Exist?

When do Averages Exist?

Chapter:
(p.195) 19 When do Averages Exist?
Source:
Chaos and Fractals
Author(s):

David P. Feldman

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199566433.003.0020

Since fractals are self-similar, it is often not useful to describe them in terms of an average size. Stating an average size does not capture what is interesting or noteworthy about the shape of a fractal object. This chapter considers a situation in which stating an average property is useless and mathematically ill-defined, and which shows that fractals are not just geometric objects but can also be used to describe processes that unfold in time. As a first example, it presents a simple game of tossing a coin, to illustrate what it means for something to possess an average. It then describes the St. Petersburg paradox, also known as the St. Petersburg game or lottery, and calculates the average winnings for this game.

Keywords:   fractals, average, tossing a coin, St. Petersburg paradox, average size, St. Petersburg game

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