This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Suppose P ( E  F ) = P ( E ), i.e., knowing F doesnt help in predicting E . Then E and F are indepen dent. What we have said is that in this case P ( E  F ) = P ( E F ) P ( F ) = P ( E ) , or P ( E F ) = P ( E ) P ( F ). We use the latter equation as a definition: We say E and F are independent if P ( E F ) = P ( E ) P ( F ) . Example. Suppose you flip two coins. The outcome of heads on the second is independent of the outcome of tails on the first. To be more precise, if A is tails for the first coin and B is heads for the second, and we assume we have fair coins (although this is not necessary), we have P ( A B ) = 1 4 = 1 2 1 2 = P ( A ) P ( B ). Example. Suppose you draw a card from an ordinary deck. Let E be you drew an ace, F be that you drew a spade. Here 1 52 = P ( E F ) = 1 13 1 4 = P ( E ) P ( F ). Proposition 3.1. If E and F are independent, then E and F c are independent. Proof. P ( E F c ) = P ( E ) P ( E F ) = P ( E ) P ( E ) P ( F ) = P ( E )[1 P ( F )] = P ( E ) P ( F c ) ....
View
Full
Document
 Fall '06
 SCHWAGER

Click to edit the document details