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LEC_EAS305_F10_0913

LEC_EAS305_F10_0913 - Conditional Probability Bayes Theorem...

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Conditional Probability Bayes’ Theorem Fall 2010 EAS 305 Lecture Notes Prof. Jun Zhuang University at Buffalo, State University of New York September 13, 15, ... 2010 Prof. Jun Zhuang Fall 2010 EAS 305 Lecture Notes Page 1 of 26
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Conditional Probability Bayes’ Theorem Agenda for Today 1 Conditional Probability Definition and Properties Independence General Definition 2 Bayes’ Theorem Partition Theorem Examples Prof. Jun Zhuang Fall 2010 EAS 305 Lecture Notes Page 2 of 26
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Conditional Probability Bayes’ Theorem Definition and Properties Independence General Definition Example Example: Die. A = { 2 , 4 , 6 } , B = { 1 , 2 , 3 , 4 , 5 } . So Pr( A ) = 1 / 2, Pr( B ) = 5 / 6. Suppose we know that B occurs. Then the prob of A “given” B is Pr( A | B ) = 2 5 = | A B | | B | So the prob of A depends on the info that you have! The info that B occurs allows us to regard B as a new, restricted sample space. And. . . Pr( A | B ) = | A B | | B | = | A B | / | S | | B | / | S | = Pr( A B ) Pr( B ) . Prof. Jun Zhuang Fall 2010 EAS 305 Lecture Notes Page 3 of 26
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Conditional Probability Bayes’ Theorem Definition and Properties Independence General Definition Definition: If Pr( B ) > 0, the conditional prob of A given B is Pr( A | B ) Pr( A B ) / Pr( B ). Remarks: If A and B are disjoint, then Pr( A | B ) = 0. (If B occurs, there’s no chance that A can also occur.) What happens if Pr( B ) = 0? Don’t worry! In this case, makes no sense to consider Pr( A | B ). Prof. Jun Zhuang Fall 2010 EAS 305 Lecture Notes Page 4 of 26
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Conditional Probability Bayes’ Theorem Definition and Properties Independence General Definition Example: Toss 2 dice and take the sum. A : odd toss = { 3 , 5 , 7 , 9 , 11 } B : { 2 , 3 } Pr( A ) = Pr(3) + · · · + Pr(11) = 2 36 + 4 36 + · · · + 2 36 = 1 2 . Pr( B ) = 1 36 + 2 36 = 1 12 . Pr( A | B ) = Pr( A B ) Pr( B ) = Pr(3) Pr( B ) = 2 / 36 1 / 12 = 2 / 3 . Prof. Jun Zhuang Fall 2010 EAS 305 Lecture Notes Page 5 of 26
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Conditional Probability Bayes’ Theorem Definition and Properties Independence General Definition Properties — analogous to Axioms of probability 1 0 Pr( A | B ) 1. 2 Pr( S | B ) = 1. 3 A 1 A 2 = ∅ ⇒ Pr( A 1 A 2 | B ) = Pr( A 1 | B ) + Pr( A 2 | B ). 4 If A 1 , A 2 , . . . are all disjoint, then Pr [ i =1 A i B = X i =1 Pr( A i | B ) . Prof. Jun Zhuang Fall 2010 EAS 305 Lecture Notes Page 6 of 26
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Conditional Probability Bayes’ Theorem Definition and Properties Independence General Definition Independence — Any unrelated events are independent Definition: A and B are independent iff Pr( A B ) = Pr( A )Pr( B ).
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