By Kardi Teknomo, PhD .
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The usual way to compute timevariance is given by equation (2). I put here again for your convenient:
_{ } (2)
Changing _{ } to _{ } makes equation (2) becomes
_{ }
or
_{ } (a)
Putting out the last term from equation (2) gives
_{ }
Since then
_{ } (b) ( why? )
Equation (3) can be rearranged into
_{ } (c)
The second term of the right hand side of
equation (b)
can be manipulated
as
_{ }
_{ }
Put back the second term into equation (b) and replace the third term with equation (a) becomes
_{ }
_{ } (proved)
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These tutorial is copyrighted .
Preferable reference for this tutorial is
Teknomo, Kardi. (2006) Recursive Average and Variance.
http://people.revoledu.com/kardi/tutorial/RecursiveStatistic/index.html