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Probability Numbers
For many statisticians the concept of “the probability that an event occurs” is ultimately
rooted in the interpretation of an event as an outcome of an experiment, others would
interpret the concept as a subjective degree of belief that the event occurs.
This
difference in interpretation has been the source of great debate between Frequentists (the
former group) and Bayesians (the latter group) within the statistics profession.
This
philosophical debate ultimately influences the way that empirical results are obtained and
interpreted, but does not affect the use of probability ideas employed in describing the
distribution of happiness in a society.
Hence we shall proceed under the Frequentist
interpretation whilst acknowledging the existing of an alternative.
At the heart of the frequentists view of probability is the notion of an experiment.
A
sufficiently general definition of the term “Experiment” permits an almost universal
application of the probability concept.
Essentially a procedure must possess two
properties to be eligible as an experiment.
1)
It should be notionally repeatable an infinite number of times with a well
defined common set of possible outcomes each time the procedure takes
place.
2)
There should be uncertainty as to which outcome will occur before the
procedure takes place.
Probability Theory is best understood by using ideas from set theory in its description.
The appendix provides an outline of the basic set theory ideas that are used.
The set of mutually exclusive (having nothing in common) and exhaustive (a complete
list of) possible
Basic Outcomes
(denoted o
i
) of an experiment is usually called
The
Sample Space
(denoted S).
An
Event
is defined as any subset of this sample space,
including the empty set and the sample space itself, generally an event is denoted by an
upper case letter say A, A
c
, the complement of A, then corresponds to A not happening.
An event is said to have occurred when any one of the basic outcomes in its defining
subset is realized.
Thus, on executing the procedure, the empty or null set is the event
“no outcome occurs” which is certain not to happen, similarly the sample space set (the
universal set in set theoretic terms) is the event “an outcome occurs” which is certain to
happen.
Given outcome uncertainty there is obviously event uncertainty ranging in
degree between the empty set and the sample space.
Numbers attached to events reflecting the degree of certainty with which they occur, and
which in turn obey certain coherency axioms, are called “Probabilities.”
The coherency
axioms are simple yet remarkably powerful, they provide the basis for all probability
theory regardless of its objective or subjective foundations.
Denoting the occurrence of
the i’th basic outcomes as o
i
and the probability of it happening as P(o
i
) and denoting
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View Full Document events by upper case letters (with S and
∅
respectively reserved for the sample and
empty sets) the coherency axioms are as follows:
1)
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This note was uploaded on 03/23/2011 for the course ECONOMICS 209 taught by Professor Kambourov during the Spring '11 term at University of Toronto Toronto.
 Spring '11
 Kambourov

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