W8-slides3 - Lecture 24 Outline and Examples Conditional...

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Unformatted text preview: Lecture 24- Outline and Examples Conditional distributions (Ross § 6.4 and § 6.5) Conditional probabilities Recall, the conditional probability of event A occurring given that event B has occurred is P ( A | B ) = P ( A ∩ B ) P ( B ) provided P ( B ) 6 = 0 i.e. the event on which we are conditioning has non zero probability. Using the previous definition of conditional probabilities of events we have P ( Y ∈ A | X = x ) = P ( Y ∈ A , X = x ) P ( X = x ) . I cannot use this formula in the continuous random variable case since P ( X = x ) = 0 in the continuous case! Therefore I need to define conditional probabilities of random variables differently. We define it as P ( Y ∈ A | X = x ) = Z A f Y | X ( y | x ) dy . What should this function f Y | X ( y | x ) be? Conditional distributions Discrete case: The conditional probability mass function of Y given that X = x is p Y | X ( y | x ) := P ( Y = y | X = x ) = P ( X = x , Y = y ) P ( X = x ) = p X , Y ( x , y ) p X ( x ) . (defined only when p X ( x ) 6 = 0.) P ( Y ∈ A | X = x ) = X y ∈ A p Y | X ( y | x ) . Continuous case: The conditional probability density of Y given X = x is f Y | X ( y | x ) = f X , Y ( x , y ) f X ( x ) ....
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This note was uploaded on 06/05/2010 for the course STAT PStat 120a taught by Professor Rohinikumar during the Spring '10 term at UCSB.

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W8-slides3 - Lecture 24 Outline and Examples Conditional...

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