Lecture5

Lecture5 - Conditional Probability(cont Independent Events...

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Conditional Probability (cont...) 10/06/2005

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Independent Events Two events E and F are independent if both E and F have positive probability and if P ( E | F ) = P ( E ) , and P ( F | E ) = P ( F ) . 1
Theorem. If P ( E ) > 0 and P ( F ) > 0 , then E and F are inde- pendent if and only if P ( E F ) = P ( E ) P ( F ) . 2

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Example Suppose that we have a coin which comes up heads with proba- bility p , and tails with probability q . Now suppose that this coin is tossed twice. Let E be the event that heads turns up on the first toss and F the event that tails turns up on the second toss. Are these independent? What if A is the event“the first toss is a head”and B is the event “the two outcomes are the same”? What about I and J , where I is the event “heads on the first toss”and J is the event“two heads turn up.” 3
A set of events { A 1 , A 2 , . . . , A n } is said to be mutually in- dependent if for any subset { A i , A j , . . . , A m } of these events we have P ( A i A j ∩ · · · ∩ A m ) = P ( A i ) P ( A j ) · · · P ( A m ) , or equivalently, if for any sequence ¯ A 1 , ¯ A 2 , . . . , ¯ A n with ¯ A j = A j or ˜ A j , P ( ¯ A 1 ¯ A 2 ∩ · · · ∩ ¯ A n ) = P ( ¯ A 1 ) P ( ¯ A 2 ) · · · P ( ¯ A n ) . 4

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Joint Distribution Functions If we have several random variables X 1 , X 2 , . . . , X n which corre- spond to a given experiment, then we can consider the joint random variable ¯ X = ( X 1 , X 2 , . . . , X n ) defined by taking an outcome ω of the experiment, and writing, as an n -tuple, the corresponding n outcomes for the random variables X 1 , X 2 , . . . , X n . Thus, if the random variable X i has, as its set of possible outcomes the set R i ,
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