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Conditioning and Martingales

  • Vivek S. Borkar
Part of the Universitext book series (UTX)

Abstract

Let (Ω,F, P) be a probability space and A, BF events such that P(B) > 0. In this section we seek to formalize the intuitive content of the phrase “probability of A given (or, equivalently, “conditioned on”) B”. The words “given B ” imply prior knowledge of the fact that the sample point ω lies in B. It is then natural to consider the reduced probability space (Ω, Fb, Pb) with ΩB = B,FB = A∩B AF (the “trace σ-field”), PB(C) = P(C)/P(B) for CTB, and define the probability of A given B, denoted by P(A/B), as PB(A ∩ B) = P(AB)/P(B). The reader may convince himself of the reasonableness of this procedure by evaluating, say, the probability of two consecutive heads in four tosses of a fair coin given the knowledge that the number of heads was even, using the principle of insufficient reason.

Keywords

Conditional Expectation Dominate Convergence Theorem Polish Space Martingale Theory Real Random Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Vivek S. Borkar
    • 1
  1. 1.Department of Electrical EngineeringIndian Institute of ScienceBangaloreIndia

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