Independent Pr[A] = Pr[A | B] Definition 2: Events Events A and B - - PowerPoint PPT Presentation

Albert R Meyer, May 3, 2013 indep

Mathematics for Computer Science

MIT 6.042J/18.062J

Independent Events

inde -events.1 p-events.1 Albert R Meyer, May 3, 2013 indep

Independent Events

Definition 1: Events A and B are independent iff

Pr[A] = Pr[A | B]

Definition 2: Events A and B are independent iff

Pr[A] ⋅ Pr[B] = Pr[A ∩ B]

inde -events.2 p-events.2 Albert R Meyer, May 3, 2013 indep

Definitions of Independence

proof of equivalence:

Pr[A] = Pr[A | B]

Pr[A ∩

iff Pr[A] = iff Pr[A] ⋅ Pr[B] = Pr[A ∩ B]

• events.3

∩ B] Pr[B]

indep-events.3 Albert R Meyer, May 3, 2013 indep

Definitions of Independence

Pr[B] ≠ 0 for Def. 1. need

• Def. 2 always works:

Pr[A}⋅Pr[B] = Pr[A∩B]

• events.4

indep-events.4

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Independence

Pr[A]⋅Pr[B] = Pr[A∩B]

symmetric in A and B so, A independent of B iff B independent of A

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Independence

Corollary: If Pr[B]= 0, then

B is independent of every event

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Independence

Corollary: If Pr[B]= 0, then

B is independent of every event – even itself.

indep-events.7 Albert R Meyer, May 3, 2013 indep-events.9

Independence

A independent of B means

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Independence

A independent of B means A is independent of whether or not B occurs:

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A independent of B iff A independent of .

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B

Lemma:

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Independence

Albert R Meyer, May 3, 2013

A independent of B iff A independent of

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B

Lemma:

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Independence

Simple proof using: Pr[A-B] = Pr[A]-Pr[A∩B]

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