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Lecture 5 - Independence &amp; Conditional Indepedence

# Lecture 5 - Independence &amp; Conditional Indepedence...

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1 INDEPENDENCE Lecture 5 ORIE3500/5500 Summer2011 Li Independence and Conditional Independence 1 Independence Deﬁnition. Two events A and B are said to be independent if P ( A B ) = P ( A ) P ( B ) . This can be written in terms of conditional probability as well. Note that the deﬁnition is equivalent to saying P ( A | B ) = P ( A ) when P ( B ) > 0. If P ( B ) = 0 then A and B are always independent. Example Does there exist a set A such that A is independent of itself? We need P ( A A ) = P ( A ) 2 which means P ( A )(1 P ( A )) = 0. Therefore, we only need P ( A ) = 0 or 1. For example, A = ϕ or A = Ω. It would be observed that the concepts of disjoint sets and independence are diﬀerent. Independence is deﬁned with the probability of that set. A collection of N events A 1 ,A 2 ,...,A n are said to be independent if any selection of r (1 < r n ) events A k 1 ,A k 2 ,...,A k r (1 k 1 < k 2 < ··· < k r n ) from the collection of n events, satisﬁes P ( A k 1 A k 2 ∩ ··· ∩ A k r ) = P ( A k 1 ) P ( A k 2 ) ...P ( A k r ) . They are said to be pairwise independent if for any

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Lecture 5 - Independence &amp; Conditional Indepedence...

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