Lecturenotes16

# Lecturenotes16 - STAT 430/510 Lecture 16 STAT 430/510...

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STAT 430/510 Lecture 16 STAT 430/510 Probability Hui Nie Lecture 16 June 24th, 2009

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STAT 430/510 Lecture 16 Review Sum of Independent Normal Random Variables Sum of Independent Poisson Random Variables Sum of Independent Binomial Random Variables Conditional Distributions: Discrete Case
STAT 430/510 Lecture 16 Conditional Distributions: Continuous Case Let X and Y be jointly continuous r.v.’s. Then for any x value for which f X ( x ) > 0, the conditional pdf of Y given X = x is f Y | X ( y | x ) = f ( x , y ) f X ( x ) , - ∞ < y < For any set A , P ( Y A | X = x ) = Z A f Y | X ( y | x ) dy = Z A f ( x , y ) f X ( x ) dy If X and Y are independent, then f Y | X ( y | x ) = f ( x , y ) f X ( x ) = f X ( x ) f Y ( y ) f X ( x ) = f Y ( y )

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STAT 430/510 Lecture 16 Example The joint density of X and Y is given by f ( x , y ) = 12 5 x ( 2 - x - y ) , 0 < x < 1 , 0 < y < 1 0 , otherwise Compute the conditional density of X given that Y = y , where 0 < y < 1. f X | Y ( x | y ) = f ( x , y ) f Y ( y ) = x ( 2 - x - y ) R 1 0 x ( 2 - x - y ) dx = 6 x ( 2 - x - y ) 4 - 3 y
STAT 430/510 Lecture 16 One Discrete R.V. and one Continuous R.V.

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