Notes
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Outline
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Probability Theory
  • 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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Approaches
  • Physical
    • Experiment (frequency ratio, apostiori)
    • Classical Definition (# of trials large)
  • Conceptual - Axiomatic
    • Axioms and Reasoning (heavy math)
  • Prediction
    • Symmetry (thought experiment, apriori)
    • The Sample Space
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Sample Space
  • S is the collection or set of all possible outcomes of an experiment.  It is the universal set for this experiment.
    • Discrete Sample Space:  Finite or countably infinite set.  (e.g. faces of a die, the positive integers, students in this class)
    • Continuous Sample Space: not countable.  (e.g. the real line, battery voltage in a car)
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The Probability Function
  • Definition
  • To each outcome in S we associate a non-negative number Pn = P(xn) such that :
  • 0<Pn<1 and Sum[ P(xn)] = 1
    • xn is in S
    • Sum is over all elements of S
    • {xn} is collectively exhaustive and mutually exclusive
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Truths
  • P(A) is greater than or equal to 0
  • P(S) = 1
  • P(A + B) = P(A) + P(B) - P(A*B)
  • P({}) = 0
  • Mutually Exclusive Events:
    P(A + B) = P(A) + P(B)
  • The three axioms of probability