Book contents
- Frontmatter
- Contents
- Preface to the second edition
- Preface to the first edition
- Some frequently used notations
- 1 Measure theory and probability
- Solutions for Chapter 1
- 2 Independence and conditioning
- Solutions for Chapter 2
- 3 Gaussian variables
- Solutions for Chapter 3
- 4 Distributional computations
- Solutions for Chapter 4
- 5 Convergence of random variables
- Solutions for Chapter 5
- 6 Random processes
- Solutions for Chapter 6
- Where is the notion N discussed?
- Final suggestions: how to go further?
- References
- Index
2 - Independence and conditioning
Published online by Cambridge University Press: 05 August 2012
- Frontmatter
- Contents
- Preface to the second edition
- Preface to the first edition
- Some frequently used notations
- 1 Measure theory and probability
- Solutions for Chapter 1
- 2 Independence and conditioning
- Solutions for Chapter 2
- 3 Gaussian variables
- Solutions for Chapter 3
- 4 Distributional computations
- Solutions for Chapter 4
- 5 Convergence of random variables
- Solutions for Chapter 5
- 6 Random processes
- Solutions for Chapter 6
- Where is the notion N discussed?
- Final suggestions: how to go further?
- References
- Index
Summary
“Philosophy” of this chapter
(a) A probabilistic model {(Ω, ℱ, P); (Xi)i∈I} consists of setting together in a mathematical way different sources of randomness, i.e. the r.v.s (Xi)i∈I usually have some complicated joint distribution
It is always a simplification, and thus a progress, to replace this “linked” family by an “equivalent” family (Yj)j∈J of independent random variables, where by equivalence we mean the equality of their σ-fields: σ(Xi, i ∈ I)= σ(Yj, j ∈ J) up to negligible sets.
(b) Assume that the set of indices I splits into I1 + I2, and that we know the outcomes {Xi(ω); i ∈ I1}. This modifies deeply our perception of the randomness of the system, which is now reduced to understanding the conditional law of (Xi)i∈I2, given (Xi)i∈I1. This is the main theme of D. Williams' book [64].
(c) Again, it is of great interest, even after this conditioning with respect to (Xi)i∈I1, to be able to replace the family (Xi)i∈I2 by an “equivalent” family (Yj)j∈J2, which consists of independent variables, conditionally on (Xi)i∈I1.
Note that the terms “independence” and “conditioning” come from our everyday language and are very suitable as translations of the corresponding probabilistic concepts. However, some of our exercises aim at pointing out some traps which may originate from this common language meaning.
(d) The Markov property (in a general framework) asserts the conditional independence of the “past” and “future” σ-fields given the “present” σ-field.
- Type
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- Information
- Exercises in ProbabilityA Guided Tour from Measure Theory to Random Processes, via Conditioning, pp. 27 - 47Publisher: Cambridge University PressPrint publication year: 2012