131A_1_Lecture_5-1_Winter_2012

131A_1_Lecture_5-1_Winter_2012 - EE 131A Probability...

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UCLA EE131A (KY) 1 EE 131A Probability Professor Kung Yao Electrical Engineering Department University of California, Los Angeles Lecture 5-1
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UCLA EE131A (KY) 2 Independence of Events (1) Consider two events A and B . If the occurrence of event B does not affect the probability of event A , then A and B are independent . Formally, the independence of A and B is given by P( A ) = P( A | B ) = P( A B ) / P( B ) . (1) Equivalently, P( A ) P( B ) = P( A B ) . (2) But P( A B ) = P( B | A ) P( A ) . (3) Equating the r.h.s. of (2) to the l.h.s. of (3), yields P( A ) P( B ) = P( B | A ) P( A ) or P( B ) = P( B | A ) . (4)
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UCLA EE131A (KY) 3 Independence of Events (2) The concept of independence of two events is intuitively clear in some simple experiences and not so obvious in more complex situations. Consider a random experiment in which we flip a coin and also toss a die, from the physics of the situation, we believe the output of the coin is independent of the outcome of the die (to be confirmed in Ex. 1.) On the other hand, the av. weight of the corn at harvest time in Iowa and the av. amount of rainfall in the last three months before harvesting in Iowa are probably not independent (needs a lot of statistical data to confirm this conjecture).
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UCLA EE131A (KY) 4 Independence of Events (3) From (1), (2), and (4) of page 2, formally the independence of
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131A_1_Lecture_5-1_Winter_2012 - EE 131A Probability...

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