# High order joint moments calculation

1. Dec 7, 2012

### Wu Xiaobin

Recently I have been working on classical Gaussian electrical field and I come through this joint
moments calculation.
Suppose we got the joint density function as:
$p(s_i,s_j)=\frac{1}{2\pi d}\exp{[\frac{1}{2d}(<s_i>s_i+<s_j>s_j)]}K_0(\frac{1}{2d}\sqrt{<I>^2-s_k^2}\sqrt{s_i^2+s_j^2})$
$<I>,<s_i>,<s_j>,<s_k>$ are the known mean of $I,s_i,s_j,s_k$.
the high order moment of $<s_i^n>$ can be calculated as:
$<s_i^n>=\frac{n!}{2^{n+1}\sqrt{<I>^2-<s_j>^2-<s_k>^2}}[(\sqrt{<I>^2-<s_j>^2-<s_k>^2}+<s_i>)^{n+1}+(-1)^n(\sqrt{<I>^2-<s_j>^2-<s_k>^2}-<s_i>)^{n+1}]$
with the above result, The paper I referred to give the following conclusion which I can't catch up with:
$<s_i^ns_j^m>=(\frac{2d}{<s_j>})^m[\frac{\partial^m<s_i^n(x,s_j)>}{\partial x^m}]_{x=1}$
where
$<s_i^n(x,s_j)>$
is given by multiplying $<s_j>^2$, in the square brackets in $<s_i^n>$
Can anyone tell me why the author calculates the joint moments in that way.

Look forward to your reply
Sincerely yours
Jacky Wu

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• ###### Statistics of Stokes variables for correlated Gaussian Fields.pdf
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