# 06_29ans - STAT 410 Examples for Summer 2009 2.3 1...

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STAT 410 Examples for 06/29/2009 Summer 2009 2.3 Conditional Distributions and Expectations. 1. Consider the following joint probability distribution p ( x , y ) of two random variables X and Y: y x 0 1 2 p X ( x ) 1 0.15 0.15 0 0.30 2 0.15 0.35 0.20 0.70 p Y ( y ) 0.30 0.50 0.20 a) Find the conditional probability distributions p X | Y ( x | y ) = ( ( 29 y p y x p , Y of X given Y = y , conditional expectation E ( X | Y = y ) of X given Y = y , conditional variance Var ( X | Y = y ) of X given Y = y , E ( E ( X | Y ) ) , and Var ( E ( X | Y ) ) . x p X | Y ( x | 0 ) x p X | Y ( x | 1 ) x p X | Y ( x | 2 ) 1 0.15 / 0.30 = 0.50 1 0.15 / 0.50 = 0.30 1 0.00 / 0.20 = 0.00 2 0.15 / 0.30 = 0.50 2 0.35 / 0.50 = 0.70 2 0.20 / 0.20 = 1.00 E ( X | Y = 0 ) = 1.5 E ( X | Y = 1 ) = 1.7 E ( X | Y = 2 ) = 2.0 Var ( X | Y = 0 ) = 0.25 Var ( X | Y = 1 ) = 0.21 Var ( X | Y = 2 ) = 0.00 Def E ( X | Y = y ) = x x P ( X = x | Y = y ) = x x p X | Y ( x | y ) Denote by E ( X | Y ) that function of the random variable Y whose value at Y = y is E ( X | Y = y ) . Note that E ( X | Y ) is itself a random variable, it depends on the ( random ) value of Y that occurs.

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E ( a 1 X 1 + a 2 X 2 | Y ) = a 1 E ( X 1 | Y ) + a 2 E ( X 2 | Y ) E [ g ( Y ) | Y ] = g ( Y ) E ( E ( X | Y ) ) = E ( X ) E [ E ( X | Y ) | Y ] = E ( X | Y ) E [ g ( Y ) X | Y ] = g ( Y ) E ( X | Y ) E [ g ( Y ) E ( X | Y ) ] = E [ g ( Y ) X ] Def Var ( X | Y ) ) = E [ ( X – E ( X | Y ) ) 2 | Y ] = E ( X 2 | Y ) [ E ( X | Y ) ] 2 y E ( X | Y = y ) p Y ( y ) Var ( X | Y = y ) p Y ( y ) 0 1.5 0.30 0.25 0.30 1 1.7 0.50 0.21 0.50 2 2.0 0.20 0.00 0.20 E ( E ( X | Y ) ) = 1.7 = E ( X ) E ( Var ( X | Y ) ) = 0.18 Var ( E ( X | Y ) ) = 0.03 < 0.21 = Var ( X ) . 0.21 = 0.03 + 0.18. Theorem E ( E ( X | Y ) ) = E ( X ) Var ( E ( X | Y ) ) Var ( X ) Furthermore, Var ( X ) = Var ( E ( X | Y ) ) + E ( Var ( X | Y ) )
b) Find the conditional probability distributions p Y | X ( y | x ) = ( ( 29 x p y x p , X of Y given X = x , conditional expectation E ( Y | X = x ) of Y given X = x ,

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