Recently I have been working on classical Gaussian electrical field and I come through this joint(adsbygoogle = window.adsbygoogle || []).push({});

moments calculation.

Suppose we got the joint density function as:

[itex]

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})

[/itex]

[itex]<I>,<s_i>,<s_j>,<s_k>[/itex] are the known mean of [itex]I,s_i,s_j,s_k[/itex].

the high order moment of [itex]<s_i^n>[/itex] can be calculated as:

[itex]

<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}]

[/itex]

with the above result, The paper I referred to give the following conclusion which I can't catch up with:

[itex]

<s_i^ns_j^m>=(\frac{2d}{<s_j>})^m[\frac{\partial^m<s_i^n(x,s_j)>}{\partial x^m}]_{x=1}

[/itex]

where

[itex]

<s_i^n(x,s_j)>

[/itex]

is given by multiplying [itex]<s_j>^2[/itex], in the square brackets in [itex] <s_i^n>[/itex]

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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# High order joint moments calculation

Can you offer guidance or do you also need help?

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