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Unformatted text preview: are independent if they are pairwise independent and if
P (ABC ) = P (A)P (B )P (C ).
Suppose A, B , C are independent. Then A (or Ac ) is independent of any event that can be
made from B and C by set operations. For example, A is independent of BC because P (A(BC )) =
P (ABC ) = P (A)P (B )P (C ) = P (A)P (BC ). For a somewhat more complicated example, here’s a
proof that A is independent of B ∪ C :
P (A(B ∪ C )) = P (AB ) + P (AC ) − P (ABC )
= P (A)[P (B ) + P (C ) − P (B )P (C )]
= P (A)P (B ∪ C ).
If the three events A, B , and C have to do with three physically separated parts of a probability experiment, then we would expect them to be independent. But three events could happen to be independent even if they are not physically separated. The deﬁnition of independence
for three events involves four equalities–one for each pairwise independence, and the ﬁnal one:
P (A)P (B )P (C ) = P (A)P (B )P (C ).
Finally, we give a deﬁnition of independence for any ﬁnite collection of events, which generalizes
the above d...
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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 Institute of Technology.
- Spring '08
- The Land