ECE313.Lecture19

# ECE313.Lecture19 - ECE 313 Probability with Engineering...

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System Reliability 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 19 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 38 Independent events Definition: Events A and B defined on an experiment are said to be (stochastically) mutually independent if P(A B) = P(A)P(B) Sometimes people say “A is independent of B” instead, but independence is mutual: A is independent of B if and only if B is independent of A
ECE 313 - Lecture 19 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 38 What’s with the “stochastically”? P(A B) = P(A)P(B) for stochastically independent events If we believe that events A and B are physically independent, then we insist that the probability measure must assign probabilities to events in such a way that this equality holds But, equality can hold even for events that are provably physically dependent… such events are stochastically independent

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ECE 313 - Lecture 19 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 38 Physical stochastic Physical independence is, in essence, a property of the events themselves We believe that events A and B are physically independent and express this independence via P(AB) = P(A)P(B) Stochastic independence is a property of the probability measure Stochastic independence does not necessarily mean that the events are physically independent
ECE 313 - Lecture 19 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 38 Consequences of independence If A and B are mutually independent events, then P(AB) = P(A)P(B) This is equivalent to each of the following P(AB c ) = P(A)P(B c ) P(A c B) = P(A c )P(B) P(A c B c ) = P(A c )P(B c ) In other words, A and B c are mutually independent, as are A c and B, and as for A c and B c , why, they are independent too!

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ECE 313 - Lecture 19 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 38 Mutually is as mutually does… If A and B are mutually independent events, then P(AB) = P(A)P(B) If A and B are mutually exclusive events, then P(AB) = 0 Mutually exclusive events cannot be mutually independent Mutually independent events cannot be mutually exclusive Except in trivial cases: P(A) or P(B) is 0
© 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 38 Conditional = unconditional If A and B are mutually independent events, then P(AB) = P(A)P(B) If P(A) > 0, we get that P(B | A) = P(AB)/P(A) = P(B) The conditional probability of B given A is the same as the unconditional probability! Knowing that A occurred does not cause

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ECE313.Lecture19 - ECE 313 Probability with Engineering...

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