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ConditionalExpectation

# ConditionalExpectation - f x | y = f x y/f y is the...

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STAT 333 Conditional Expectation Basic Properties of Conditional Expectation: 1. linearity E ( n j =1 a j X j | Y = y ) = n j =1 a j E ( X j | Y = y ). 2. double averaging E ( X ) = E ( E ( X | Y )) where the outer expectation on the right hand side is averaged over the values of Y . 3. substitution E ( h ( X, Y ) | Y = y ) = E ( h ( X, y ) | Y = y ) 4. independence E ( X | Y ) = E ( X ) if X, Y are independent. The above properties hold for all random variables X and Y . Most problems involving conditional expectation can be solved by applying the above principles appropriately, sometimes in conjunction with conditional information specific to the problem. In addition we have the following: If both X and Y are discrete, then E ( X | Y = y ) = x x P ( X = x | Y = y ) . If both X and Y are continuous, then E ( X | Y = y ) = x f ( x | y )
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Unformatted text preview: f ( x | y ) = f ( x, y ) /f ( y ) is the conditional p.d.f. of X given Y . • If X is discrete and Y is continuous, then we formally write E ( X ) = ² E ( X | Y = y ) f ( y ) dy and evaluate the integral by applying properties 1., 3., 4. in conjunction with any known conditional information. In particular, if X = I A (an indicator variable) then E ( I A | Y = y ) = P ( A | Y = y ) and so P ( A ) = E ( I A ) = ² P ( A | Y = y ) f ( y ) dy where the integral is evaluated using properties of conditional probability analogous to those of conditional expectation....
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