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Science Forum Index » Statistics - Math Forum » expectation of multivariate normal
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| Kaushik |
 Posted: Wed Jan 10, 2007 1:02 am |
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hello all,
i'm trying to find if there is any known result for this expectation.
suppose x \in R^{n} follows normal distribution with mean vector m and
variance co variance matrix V
then V= E[(x-m)'(x-m)]
i'm looking for the quantity E[(x-m)'A(x-m)]
where A is a n by n square matrix, specifically A=[V+V^{-1}]^{2}
is there any known result for this quantity in terms of V and trace of
the
matrix A?
thanks |
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| David Jones |
| Posted: Wed Jan 10, 2007 6:14 am |
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Kaushik wrote:
Quote: hello all,
i'm trying to find if there is any known result for this
expectation.
suppose x \in R^{n} follows normal distribution with mean vector m
and
variance co variance matrix V
then V= E[(x-m)'(x-m)]
You probably mean V= E[(x-m)(x-m)']
Quote: i'm looking for the quantity E[(x-m)'A(x-m)]
where A is a n by n square matrix, specifically A=[V+V^{-1}]^{2}
is there any known result for this quantity in terms of V and trace
of
the
matrix A?
thanks
The first steps are fairly standard ...
(x-m)'A(x-m) =trace{(x-m)'A(x-m)} = trace{A(x-m)(x-m)'}
so ...
E[(x-m)'A(x-m)] =E[trace{(x-m)'A(x-m)}]
= E[trace{A(x-m)(x-m)'}]
= trace{AV}
... then simplify the expression in terms of the trace of powers of
V.
David Jones |
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