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Say i have $n$ variables with variances $V_1,V_2,...V_n$. The sum of the variables will have a variance of $V=V_1+V_2+..V_n$ .Now if i am given N total simulations to reduce the variance V, how do i proportion the simulations between the variables to get smallest total variance?

Objective: Minimize $(V_1/N_1+V_2/N_2+...V_n/N_n)$ where N = $N_1+N_2...N_n$.

It seems that proportionaing by $sqrt(V_i)$ seems to give the smallest value and I have proved it for the case of 2 variables. But I can't do it for three or more variables. Can anyone help?

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  • $\begingroup$ Welcome to MSE! It really helps readability to format questions using MathJax (see FAQ). Regards $\endgroup$ – Amzoti Apr 11 '13 at 1:51
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There'll be a neat, 'elementary' AM-GM type way to do it, but personally I find Lagrange multipliers make easy work of constraint equations like this.

The idea is to have $F = \sum V_i/N_i -\lambda(\sum N_i - N)$ satisfy $\partial F/\partial N_i = 0$ for each $i$. This should give you the answer you are looking for!

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  • $\begingroup$ That is excellent! Thanks a lot. $\endgroup$ – danny Apr 11 '13 at 2:01
  • $\begingroup$ No problem! As Amzoti suggested, try to use LaTeX in your questions in future! Also, it's conventional to 'upvote' answers you think contribute usefully by clicking the up arrow above the number found at the top left of each answer. $\endgroup$ – Sharkos Apr 11 '13 at 2:09

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