ECE313.Lecture14

# ECE313.Lecture14 - ECE 313 Probability with Engineering...

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Bayes’ Formula 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 14 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 38 Conditional probability Given that event A of probability P(A) > 0 occurred, the conditional probability of B given A is denoted by P(B | A) and deFned as P(B | A) = P(AB) P(A) P(B | A) can be larger than, smaller than, or the same as P(B) Conditional probabilities satisfy the axioms of probability theory
ECE 313 - Lecture 14 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 38 The chain rule or product rule P(B | A) = P(AB)/P(A) P(AB) = P(B | A)P(A) P(ABCD…) = P(A)P(B | A)P(C | AB)P(D | ABC)… The chain rule also applies to conditional probabilities given an event H (say) P(ABCD… | H ) = P(A | H )P(B | A H )P(C | AB H )P(D | ABC H )…

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ECE 313 - Lecture 14 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 38 The theorem of total probability I The theorem of total probability allows us to compute unconditional probabilities from conditional probabilities P(A) = P(A | B)P(B) + P(A | B c )P(B c ) P(B) = P(B | A)P(A) + P(B | A c )P(A c ) The theorem also applies to conditional probabilities P(A | C) = P(A | BC)P(B | C)+P(A | B c c
ECE 313 - Lecture 14 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 38 The theorem of total probability II Given a countable partition A 1 , A 2 , … A n , … of the sample space, P(B) = P(B | A 1 )P(A 1 ) + P(B | A 2 )P(A 2 ) + … + P(B | A n )P(A n ) + … P(B | C) = P(B | A 1 C)P(A 1 | C) + P(B | A 2 C)P(A 2 | C) + … + P(B | A C)P(A | C) + …

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ECE 313 - Lecture 14 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 38 Checking the answers min P(B | A i ) ≤ P(B) ≤ max P(B | A j ) In particular, P(B) ≥ min{P(B | A), P(B | A c )} P(B) ≤ max{P(B | A), P(B | A c )} Equality holds if and only if P(A) = 0 or
ECE 313 - Lecture 14 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 38 Why is all this stuff important? The chain rule or product rule allows us to compute a joint probability (probability of an intersection) as the product of various conditional probabilities The theorem of total probability allows us to Fnd an unconditional probability from conditional probabilities Results are very important and very useful tools in probabilistic analyses

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ECE 313 - Lecture 14 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 8 of 38 Conditional pmf of X The pmf of a discrete random variable X tells us the probabilities
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ECE313.Lecture14 - ECE 313 Probability with Engineering...

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