Chapter 5. Independence

Chapter 5. Independence - 5 Independence 5.1 Introduction...

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Unformatted text preview: 5 Independence 5.1 Introduction 5.1.1 Definition. The events A 1 ,A 2 ,... are independent if for any n events A i 1 ,...,A i n selected from A 1 ,A 2 ,... , we have P ( A i 1 A i 2 A i n ) = P ( A i 1 ) P ( A i 2 ) P ( A i n ) . 5.1.2 Meaning : Take a pair of events A and B such that P ( B ) > 0. Then P ( A B ) = P ( A ) P ( B ) if and only if P ( A | B ) = P ( A ) . Thus, knowing B occurs does not affect probability of A . For a sequence of events, this means that the occurrence of some of the events does not affect the probabilities of the remaining ones. 5.1.3 Examples : outcomes of tosses of a coin; outcomes of tosses of a dice; birthdays of a group of randomly selected persons, etc. Counter-examples : daily temperatures; birthdays of a group of twins; outcomes of football matches played by Manchester United. 5.2 Bernoulli trials 5.2.1 Definition. A Bernoulli trial is an experiment with only two possible outcomes: success and failure . 23 5.2.2 A sequence of Bernoulli trials is independent if the events A i = { i th trial is a success } are independent. [Equivalently, we may replace success by failure for some of the A i s in the above definition of independence.] 5.2.3 Suppose all the n independent Bernoulli trials have the same success probability, so that...
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Chapter 5. Independence - 5 Independence 5.1 Introduction...

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