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We are dealing here with mass phenomena and
their “average” behavior. The
physical system that creates the OBSERVABLE phenomena is usually assumed to
be CONSISTENT and the observation {either over time (a sequence) or among a
simultaneous population (a SET, an ENSEMBLE)} will have statistics that are
also consistent. We need a theory
that describes and predicts these statistics. |
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Physical |
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Experiment (frequency ratio, apostiori) |
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Classical Definition (# of trials large) |
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Conceptual - Axiomatic |
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Axioms and Reasoning (heavy math) |
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Prediction |
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Symmetry (thought experiment, apriori) |
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The Sample Space |
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S is the collection or set of all possible
outcomes of an experiment. It is
the universal set for this experiment. |
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Discrete Sample Space: Finite or countably infinite set. (e.g. faces of a die, the positive integers, students in this
class) |
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Continuous Sample Space: not countable. (e.g. the real line, battery voltage in
a car) |
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Definition |
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To each outcome in S we associate a
non-negative number Pn = P(xn) such that : |
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0<Pn<1 and Sum[ P(xn)]
= 1 |
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xn is in S |
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Sum is over all elements of S |
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{xn} is collectively exhaustive and
mutually exclusive |
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P(A) is greater than or equal to 0 |
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P(S) = 1 |
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P(A + B) = P(A) + P(B) - P(A*B) |
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P({}) = 0 |
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Mutually Exclusive Events:
P(A + B) = P(A) + P(B) |
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The three axioms of probability |
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Notes