Conditional expectation

Jul 2010
1
0
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.
 
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