24 independence and the binomial distribution 31 the

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Unformatted text preview: on P (B |A) = P (B ) is (A) equivalent to P (AB ) = P (A)P (B ). Let’s consider the other case: P (A) = 0. Should we consider A and B to be independent? It doesn’t make sense to condition on A, but P (Ac ) = 1, so we can consider P (B |Ac ) instead. It holds ( cB that P (B |Ac ) = PP A c )) = P (Ac B ) = P (B ) − P (AB ) = P (B ). Therefore, P (B |Ac ) = P (B ). That (A is, if P (A) = 0, knowledge that A is not true does not affect the probability of B. So it is natural to consider A to be independent of B. These observations motivate the following definition, which has the advantage of applying whether or not P (A) = 0 : Definition 2.4.1 Event A is independent of event B if P (AB ) = P (A)P (B ). Note that the condition in the definition of independence is symmetric in A and B. Therefore, A is independent of B if and only if B is independent of A. Another commonly used terminology for these two equivalent relations is to say that A and B are mutually independent. Here, “mutually” means that independence is a property of the tw...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

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