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Lect6_2700_s09

# Lect6_2700_s09 - ENGRD 2700 Basic Engineering Probability...

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Independence. Random Variables. Discrete RV’s pmf Independence Bernoulli trials Binomial distribution Expectation Title Page JJ II J I Page 1 of 37 Go Back Full Screen Close Quit ENGRD 2700 Basic Engineering Probability and Statistics Lecture 6: Independence; Random Variables and their Distributions David S. Matteson School of Operations Research and Information Engineering Rhodes Hall, Cornell University Ithaca NY 14853 USA [email protected] February 4, 2009

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Independence. Random Variables. Discrete RV’s pmf Independence Bernoulli trials Binomial distribution Expectation Title Page JJ II J I Page 2 of 37 Go Back Full Screen Close Quit 1. Independence. What does it mean for 2 events to be independent? Intuition: Knowledge about likelihood of one event occurring, does not affect estimate of likelihood that the other occurs. BUT: Intuition can be wrong, immature or totally out to lunch. Some examples would fool anyone’s intuition. Need to check the formal definition. Formal Definition. Events A and B are independent in the prob- ability model ( S, A , P ) if P ( AB ) = P ( A ) P ( B ) . If P ( A ) > 0 and P ( B ) > 0 then P ( A | B ) = P ( A ) and P ( B | A ) = P ( B ) .
Independence. Random Variables. Discrete RV’s pmf Independence Bernoulli trials Binomial distribution Expectation Title Page JJ II J I Page 3 of 37 Go Back Full Screen Close Quit Note 1. This is a technical definition which requires a technical verifica- tion. 2. The definition is relative to a given model and dependent on a given P . If you change the P , you may change whether 2 events are independent. It could be that A, B are independent in ( S, A , P 1 ) BUT A, B are not independent in ( S, A , P 2 ) 3. Independence is a function of the probability measure and is some- times confused for bad reasons with disjointness which is a pure set concept.

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Independence. Random Variables. Discrete RV’s pmf Independence Bernoulli trials Binomial distribution Expectation Title Page JJ II J I Page 4 of 37 Go Back Full Screen Close Quit Example: Choose a card at random from a deck of 52. S = { 1 , . . . , 52 } . Define events A = [chosen card is spade] B = [chosen card is ace ] Should these be independent? Note P ( A ) = 13 52 = 1 4 ; P ( B ) = 4 52 = 1 13 . AB = [ace of spades] so P ( AB ) = 1 52 . Check independence condition P ( AB ) = P ( A ) P ( B ) ?? Yes (intuitive??) since 1 52 = 1 4 · 1 13 .
Independence. Random Variables. Discrete RV’s pmf Independence Bernoulli trials Binomial distribution Expectation Title Page JJ II J I Page 5 of 37 Go Back Full Screen Close Quit Definition for more than 2 sets. Although one rarely has to check this because independence can be built into models by the model builder, the general definition is worth knowing. Given events { A 1 , . . . , A n } . What does independence mean? Let I { 1 , . . . , n } be a subset of the index set. We need P \ i I A i ) = Y i I P A i ) and this must hold for all I ⊂ { 1 , . . . , n } .

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• Spring '07
• RESNICK
• Probability distribution, Probability theory, pmf Independence, Independence Bernoulli trials, Independence. Random Variables

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