rphw2solf - Stochastic Processes nctuee07f Homework 2...

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Stochastic Processes nctuee07f Homework 2 Solutions 1. The solution to this problem is put on the final page. 2. This problem is very important. It states that, when the jointly Gaussian random vectors y and z are dependent, conditioning on z can always be replaced by conditioning on another Gaussian random vector ˆ z , where ˆ z and y are statistically independent. We will use this property later this semester when discussing the recursive minimum mean squared estimator, or the so-called Kalman filter . (a) Let s , [ y T z T ] T be the ( m + r ) × 1 vector collecting y and z . In topic 3, we learn that the conditional mean E [ x | y , z ] = E [ x | s ] = m x + K xs K s - 1 ( s - m s ) , (1) where m x and m s are the mean vector of x and s , respectively. In the following, the notation m * refers to the mean vector of * . We first carry out K xs and K s - 1 . The cross-covariance matrix between x and s is an n × ( m + r ) matrix and can be written in a block matrix form as K xs = E h ( x - m x )( s - m s ) T i = [ K xy | K xz ] , where K xy and K xz are the cross-covariance matrix of [ x and y , and [ x and z , respectively. Since y and z are independent, their cross-covariance matrix K yz is a zero matrix. There- fore, the covariance matrix of s is K s = E •• y - m y z - m z · [( y - m y ) T , ( z - m z ) T ] = K y 0 0 K z . It is clear that K s - 1 = K y - 1 0 0 K z - 1 . This can be seen by directly expanding the matrix multiplication K s K s - 1 , and show the result is an identity matrix. With straightforward manipulations, we find K xs K s - 1 ( s - m s ) = [ K xy K y - 1 , K xz K z - 1 ] · y - m y z - m z = K xy K y - 1 ( y - m y ) + K xz K z - 1 ( z - m z ) = E [ x | y ] - m x + E [ x | z ] - m x . (2) Substituting (2) into (1) yields E [ x | y , z ] = E [ x | y ] + E [ x | z ] - m x , when y and z are statistically independent. (b) This part is somewhat involved, and its result is very important, as I mentioned. Intu-
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This note was uploaded on 11/28/2010 for the course EE 301 taught by Professor Gfung during the Winter '10 term at National Chiao Tung University.

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rphw2solf - Stochastic Processes nctuee07f Homework 2...

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