[Springer texts in statistics] Allan Gut - Probability a graduate course (2013, Springer)

Springer Texts in Statistics
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Allan Gut
Probability: A Graduate
Course
Second Edition
123

Allan Gut
Department of Mathematics
Uppsala University
Uppsala
Sweden
ISSN 1431-875X
ISBN 978-1-4614-4707-8 ISBN 978-1-4614-4708-5 (eBook)
DOI 10.1007/978-1-4614-4708-5
Springer New York Heidelberg Dordrecht London
Library of Congress Control Number: 2012941281
ÓSpringer Science+Business Media New York 2013
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Preface to the First Edition
Toss a symmetric coin twice. What is the probability that both tosses will yield a
head?
This is a well-known problem that anyone can solve. Namely, the probability of
a head in each toss is 1=2, so the probability of two consecutive heads is
1=21=2¼1=4.
BUT! What did we do? What is involved in the solution? What are the argu-
ments behind our computations? Why did we multiply the two halves connected
with each toss?
This is reminiscent of the centipede
1
who was asked by another animal how he
walks; he who has so many legs, in which order does he move them as he is
walking? The centipede contemplated the question for a while, but found no
answer. However, from that moment on he could no longer walk.
This book is written with the hope that we are not centipedes.
There exist two kinds of probabilists. One of them is the mathematician who
views probability theory as a purely mathematical discipline, like algebra, topol-
ogy, differential equations, and so on. The other kind views probability theory as
the mathematical modeling of random phenomena, that is with a view toward
applications, and as a companion to statistics, which aims at finding methods,
principles, and criteria in order to analyze data emanating from experiments
involving random phenomena and other observations from the real world, with the
ultimate goal of making wise decisions. I would like to think of myself as both.
What kind of a random process describes the arrival of claims at an insurance
company? Is it one process or should one rather think of different processes, such
as one for claims concerning stolen bikes and one for houses that have burnt
down? How well should the DNA sequences of an accused offender and a piece of
evidence match each other in order for a conviction? A milder version is how to
order different species in a phylogenetic tree. What are the arrival rates of
1
Cent is 100, so it means an animal with 100 legs. In Swedish the name of the animal is
tusenfoting, where ‘‘tusen’’ means 1,000 and ‘‘fot’’ is foot; thus an animal with 1,000 legs or feet.
v