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?
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!