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Background

# Background - Some Background Material for ORIE 361...

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Some Background Material for ORIE 361 Instructor: Mark E. Lewis 1 Conditional Probability Let A and B be events. Then P ( A | B ) = P ( A B ) P ( B ) . Let X and Y be discrete random variables. Then for any values x and y P ( X = x | Y = y ) = P ( X = x, Y = y ) P ( Y = y ) . Notice that X = x , Y = y , X = x, Y = y , etc. are just events (i.e., sets of outcomes). 2 Independence Two events A and B are independent if and only if P ( A B ) = P ( A ) P ( B ) . This is equivalent to saying that P ( A | B ) = P ( A ) and P ( B | A ) = B . That is, knowing A occurred does not change the probability that B occurred and vice versa. Note that events A and B (with non-zero probabilities) cannot be independent if they are mutually exclusive (i.e., if A B = then P ( A B ) = P ( A ) P ( B ) ). In general two random variables X and Y are said to be independent if for any two sets A and B , P ( X A, Y B ) = P ( X A ) P ( Y B ) . Let X and Y be discrete random variables Then X and Y are independent if and only if for all x and y , P ( X = x, Y = y ) = P ( X = x ) P ( Y = y ) . This is equivalent to saying that P ( X = x | Y = y ) = P ( X = x ) and P ( Y = y | X = x ) = P ( Y = y ) (i.e., knowing the value of

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