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
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
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 ² ² ² ² B ³ = X i =1 Pr( A i | B ) . Prof. Jun Zhuang Fall 2010 EAS 305 Lecture Notes Page 6 of 26
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(

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## This note was uploaded on 10/31/2010 for the course EE 423 taught by Professor Mitin during the Spring '10 term at SUNY Buffalo.

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LEC_EAS305_F10_0913 - Conditional Probability Bayes Theorem...

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