Solution to HW 12

# Solution to HW 12 - L-= = = =-=-=-=-= Please refer to page...

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Solution to hw 12 of the Fall 2007 ee351k 1. One can regard Y as a random variable hard to observe, and regard X as a random variable easy to observe. For each hidden Y value we have an observed X value from which we can estimate b aX Y + = . Our objective is to pick proper a, b to minimize the expectation of mean square error of the linear estimator aX+b. The expected MSE (denoted as L below) is 2 2 2 , 2 2 2 , 2 , ) ( 2 ) ( ) ( 2 ) ( 2 ) ( ] ) ( ) ( 2 [ ] ) [( b X abE X E a Y bE XY aE Y E b aX b aX Y Y E b aX Y E L X X Y Y X Y Y X Y X + + + - - = + + + - = - - = To minimize L, we have to let = = 0 0 b L a L , which means = + + - = + + - 0 2 ) ( 2 ) ( 2 0 ) ( 2 ) ( 2 ) ( 2 2 , b X aE Y E X bE X aE XY E X Y X X Y X . After simplifying, = + = + ) ( ) ( ) ( ) ( ) ( , 2 Y E b X aE XY E X bE X aE Y X Y X X X , solving for a and b, X Y Y X X Y X Y X X X Y X Y X X Y X X E X E Y E X E XY E a σ σ ρ σ σ σ ρ , 2 , 2 2 , * ) var( ) , cov( )] ( [ ) ( ) ( ) ( ) ( = = = - - = ) ( ) ( * X aE Y E b X Y - = So the least expected MSE is ) 1 )( var( 2 2 ) ( ) , cov( 2 ) var( ) var( ) ( var ] )) ( [( ] )) ( ) ( [( ] ) [( 2 , 2 2 , 2 2 , 2 , , 2 2 , 2 2 * * , 2 * , * , 2 * * , 2 * * , Y X Y Y X Y Y X Y Y X Y X X Y Y X X X Y Y X Y Y X Y X Y X X Y Y X Y X Y Y X a X a Y X a Y X
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Unformatted text preview: L-=-+ =-+ =-+ =-=---= +--=--= Please refer to page 246 in the textbook for a slightly different approach. Note that the book has X = aY+b, while here we have Y = aX+b. You could also think over what it means when Y X , takes value 0, 1 or -1. 2. Just regard X(t) as a (different) random varialbe at each time instant t a. t x x t w t X e dw e x t W P x W t P x F-+∞--= =-≥ = ≤-= ∫ ) ( ) ( ) ( ) ( b. t x t X t X e x F dx d x f-= = ) ( ) ( ) ( ) ( Note that t x < because w t x-= while > w ....
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