ECE313.Lecture03

# ECE313.Lecture03 - ECE 313 Probability with Engineering...

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The Axioms of Probability, Part I Professor Dilip V. Sarwate Department of Electrical and Computer Engineering © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign. All Rights Reserved ECE 313 Probability with Engineering Applications

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ECE 313 - Lecture 3 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 40 Review of Lecture #2 Experiments with a fnite number oF outcomes were discussed An event is a subset oF the sample space An event occurs on a trial iF the observed outcome on the trial belongs to the event On any trial, only one outcome occurs but many events occur On a trial, exactly one oF A and A c occurs
ECE 313 - Lecture 3 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 40 Review of Lecture #2 (continued) Probabilities were assigned to events using the classical equally-likely outcomes approach More general assignments of probabilities (when the outcomes are not equally likely) were discussed Disjoint unions were deFned The axioms of probability were stated and some consequences were derived

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ECE 313 - Lecture 3 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 40 Probability Axioms for Fnite spaces Probabilities are numbers assigned to events, subject to the following rules: Axiom I: P(A) ≥ 0 for all events A Axiom II: P( ) = 1 Axiom III: If events A and B are disjoint, then P(A B)= P(A) + P(B) Consequences: P( ) = 0 P(A c ) = 1 – P(A); P(A) = 1 – P(A c ) 0 ≤ P(A) ≤ 1 for all events A
ECE 313 - Lecture 3 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 40 YOU MUST REMEMBER THIS!!! 0 ≤ P(A) ≤ 1 for all events A In this course, answers such as P(A) = – 0.1 or P(A) = 1.57 lead to course grades that resemble the sixth letter of the roman alphabet

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ECE 313 - Lecture 3 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 40 Partitions For any given event A, is the disjoint union of A and A c : = A A c A and A c are said to be a partition of Sets {A, B, C, …, G} are a partition of set H if H is the disjoint union of A, B,…,G Axiom III straightforwardly generalizes to give P(H) = P(A) + P(B) + … + P(G) A B and A B c are a partition of A
ECE 313 - Lecture 3 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 40 An important partition A B and A B c are a partition of A A B and A c B are a partition of B B B c A B A B c A A c B

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ECE 313 - Lecture 3 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 8 of 40 Another partition of A B is often abbreviated to just AB AB, A c B, AB c , and A c B c are a partition of AB AB c B A B c A c B A c B c
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## This note was uploaded on 09/29/2009 for the course ECE 123 taught by Professor Mr.pil during the Spring '09 term at University of Iowa.

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ECE313.Lecture03 - ECE 313 Probability with Engineering...

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