Mutually mutuallyindependent Independent whentheprobabilitythat A - PowerPoint PPT Presentation

Mathematics for Computer Science Mutual Independence MIT 6.042J/18.062J Events A 1 , A 2 ,…,A n are Mutually mutually independent Independent when the probability that A i occurs Events Albert R Meyer, May 3, 2013 Albert R Meyer, May 3, 2013 mutualindep.1 mutualindep.1 mutualindep.2 mutualindep.2 Mutual Independence Mutual Independence Events A 1 , A 2 ,…,A n are Example: Successive coin flips H i ::= [i th flip is Heads] mutually independent What happens on the 5 th flip is when the probability that independent of what happens on the 1 st , 4 th , 7 th flip: A i occurs is unchanged by ___ Pr[H 5 ] = Pr[H 5 | H 1 ∩ H 4 ∩ H 7 ] which other ones occur. Albert R Meyer, May 3, 2013 Albert R Meyer, May 3, 2013 mutualindep.3 mutualindep.3 mutualindep.5 mutualindep.5 1

Mutual Independence Mutual Independence Events A 1 , A 2 ,…,A n are Events A 1 , A 2 ,…,A n are mutually independent mutually independent when when Pr[A i ∩ A j ∩ � ∩ A m ] = Pr[A i ] = Pr[A i |A j ∩ A k ∩ � ∩ A m ] (i ≠ j,k, … ,m) Pr[A i ] ⋅ Pr[A j ] � Pr[A m ] Albert R Meyer, May 3, 2013 Albert R Meyer, May 3, 2013 mutualinde mutualindep.7 .7 mutualinde mutualindep.8 .8 Pairwise Independence Pairwise Independence Example: Flip a fair coin twice Example: Flip a fair coin twice H 1 ::= [Head on 1 st flip] O is independent of H 1 : H 2 ::= [Head on 2 nd flip] O = {HT,TH}, Pr[O] = 1/2 O ::= [Odd # Heads] O ∩ H = {HT}, Pr[{HT}] = 1 / 4 Claim: O is independent of H 1 1 Pr[O ∩ H ] = 1/ 4 = Pr[O] ⋅ Pr[H ] 1 1 Albert R Meyer, May 3, 2013 Albert R Meyer, May 3, 2013 mutualinde mutualindep.10 .10 mutualinde mutualindep.11 .11 2

k-way Independence Not Mutually Independent Example: Flip a fair coin twice Example: Flip a fair coin k times H i ::= [Head on i th flip] But O, H 1 , H 2 not mutually O ::= [Odd # Heads] independent: Claim: Any set of k of these Pr[O|H ∩ H ] = 0 ≠ Pr[O] events are mutually independent, 1 2 but all k+1 of them are not. Albert R Meyer, May 3, 2013 Albert R Meyer, May 3, 2013 mutualinde mutualindep.12 .12 mutualinde mutualindep.14 .14 k-way Independence k-way Independence Events A 1 , A 2 , ... are Events A 1 , A 2 , ... are k-way independent k-way independent iff any k of them are iff any k of them are mutually independent. mutually independent. O, H 1 , … , H k are k-way, Pairwise = 2-way not (k+1)-way independent Albert R Meyer, May 3, 2013 Albert R Meyer, May 3, 2013 mutualinde mutualindep.15 .15 mutualinde mutualindep.16 .16 3

Mutual Independence Events A 1 , A 2 ,…,A n are mutually independent when they are n-way independent 2 n -(n+1) equations to check! Albert R Meyer, May 3, 2013 mutualinde mutualindep.17 .17 4

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