ECE313.Lecture18

# ECE313.Lecture18 - ECE 313 Probability with Engineering Applications Independent Events Professor Dilip V Sarwate Department of Electrical and

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Independent Events 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 18 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 45 What is independence? Repeated independent trials The outcome of any trial of the experiment does not influence or affect the outcome of any other trial The trials are said to be physically independent Physical independence is a belief It cannot be proved that the trials are independent; we can only believe
ECE 313 - Lecture 18 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 45 Simple vs Compound Experiments Consider a simple experiment with sample space = {a 1 , a 2 , … } The result of repeated independent trials of this experiment is a sequence or vector of outcomes, say, (a 5 , a 2 , a 7 , a 9 , a 1 , … ) This vector is regarded as the outcome of a compound experiment with sample space × × × Simple experiments are sub experiments

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ECE 313 - Lecture 18 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 45 Events on Compound Experiments The outcome of a compound experiment is a sequence or vector of outcomes of the form (a 5 , a 2 , a 7 , a 9 , a 1 , … ) The simple event A occurred on i-th sub experiment if the i-th outcome in this sequence is a member of the event A ⊂ Ω The compound event (A, B, C, A c , …) occurred if a 5 A, a 2 B, a 7 C, a 9 A c , …
ECE 313 - Lecture 18 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 45 Declaration of Independence The belief in independence is reflected in the assignment of probabilities to the events of the compound experiment If the trials are (believed to be) independent, then we set P(A, B, C, A c , …) = P(A)P(B)P(C)P(A c )… Both A and A c cannot occur on the same trial of the simple experiment: here they are occurring on different sub experiments

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ECE 313 - Lecture 18 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 45 What is the event A? We defined the event A on the sample space of the simple experiment The occurrence of A on the i-th trial can be viewed as an event A i defined on the compound experiment Which outcomes of the compound experiment comprise A i ? All outcomes of the form ( , , … , x, , , … ) where x A and means “don’t care”
ECE 313 - Lecture 18 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 45 Independence of the A i Consider arbitrary events A i and B j defined on the compound experiment; i ≠ j B = A and even B = A c are acceptable choices as long as i ≠ j Because of the physical independence of the subexperiments, we have that P(A i B j ) = P(A i )P(B j ) More generally, P(A 1 B 2 C 3 …) = P(A 1 )P(B 2 )P(C 3 ) …

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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.Lecture18 - ECE 313 Probability with Engineering Applications Independent Events Professor Dilip V Sarwate Department of Electrical and

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