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Math Help - Conditional expectation

  1. #1
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    Conditional expectation

    Hi,
    I am looking for a closed-form solution for the following conditional expectation:

    E{ a(k-1) | c(k) } = E{ a(k-1) | a(k)+b(k) }

    where:

    k - discrete index
    a - discrete sequence, Laplacian distributed, zero-mean,
    b - discrete sequence, Gaussian distributed, zero-mean,
    a and b are statistically independent

    I found out that E{a(n)|a(n-1)=x}=E{a(n-1)|a(n)=x}=E{a(n)*a(n-1)}/E{a(n)^2}*x.

    Experimentally I figured out that
    E{a(n-1)|c(n)}=E{a(n)|c(n)}*E{a(n)*a(n-1)}/E{a(n)^2}
    but I couldn't prove this so far. Has anybody got an idea how to derive this equation with the assumption that E{a(n-1)|a(n)=x}=E{a(n)*a(n-1)}/E{a(n)^2}*x?

    Thanks in advance.
    Last edited by maxon; July 5th 2010 at 02:07 AM.
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