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Albert R Meyer, May 3, 2013 mutualindep.1

Mutually Independent Events

Mathematics for Computer Science

MIT 6.042J/18.062J

mutualindep.1 Albert R Meyer, May 3, 2013

Mutual Independence

Events A1, A2,…,An are

mutually independent when the probability that Ai occurs

mutualindep.2 mutualindep.2 Albert R Meyer, May 3, 2013

Mutual Independence

Events A1, A2,…,An are

mutually independent when the probability that Ai occurs is unchanged by which other ones occur.

mutualindep.3 mutualindep.3 Albert R Meyer, May 3, 2013

Mutual Independence

mutualindep.5 mutualindep.5

Example: Successive coin flips

Hi ::= [ith flip is Heads] What happens on the 5th flip is independent of what happens on the 1st, 4th, 7th flip:

Pr[H5] = Pr[H5 | H1 ∩ H4 ∩ H7

___

]

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Mutual Independence Mutual Independence

Events A1, A2,…,An are Events A1, A2,…,An are

mutually independent mutually independent when when

Pr[Ai] = Pr[Ai |Aj ∩ Ak ∩∩ A

m]

Pr[Ai ∩ Aj ∩ ∩ A

m] =

(i ≠ j,k,…,m)

Pr[Ai] ⋅ Pr[Aj]Pr[Am]

Albert R Meyer, May 3, 2013 mutualinde mutualindep.7 .7 Albert R Meyer, May 3, 2013 mutualinde mutualindep.8 .8

Pairwise Independence Pairwise Independence

Example: Flip a fair coin twice

H1 ::= [Head on 1st flip] H2 ::= [Head on 2nd flip] O ::= [Odd # Heads] Claim: O is independent of H1

Example: Flip a fair coin twice

O is independent of H1: O = {HT,TH}, Pr[O] = 1/2 O ∩ H = {HT}, Pr[{HT}] = 1 / 4

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Pr[O ∩ H ] = 1/ 4 = Pr[O] ⋅Pr[H ]

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Albert R Meyer, May 3, 2013 Albert R Meyer, May 3, 2013 mutualinde mutualindep.10 .10 mutualinde mutualindep.11 .11

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Not Mutually Independent

k-way Independence

Example: Flip a fair coin twice

But O, H1, H2 not mutually independent:

Pr[O|H ∩ H ] = 0 ≠ Pr[O]

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Albert R Meyer, May 3, 2013 mutualinde mutualindep.12 .12

k-way Independence

Events A1, A2, ... are k-way independent iff any k of them are mutually independent. Pairwise = 2-way

Example: Flip a fair coin k times

Hi ::= [Head on ith flip] O ::= [Odd # Heads] Claim: Any set of k of these

events are mutually independent, but all k+1 of them are not.

Albert R Meyer, May 3, 2013 mutualinde mutualindep.14 .14

k-way Independence

Events A1, A2, ... are k-way independent iff any k of them are mutually independent. O, H1, …, Hk are k-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

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Mutual Independence Events A1, A2,…,An are mutually independent when they are n-way independent 2n-(n+1) equations to check!

Albert R Meyer, May 3, 2013 mutualinde mutualindep.17 .17

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