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We are trying to solve the following problem. Giving a discrete random variable $V$, and a transformation function $\tilde{v}_i=\frac{v_i}{v_i+c\cdot \mathbb{E}(V)}$ we get a new random variable $\tilde{V}$.

We want to find $c$ which maximizes the variance of $\tilde{V}$

i.e. find the maximum of the function $f(c)=\sum_{i=1}^np_i\cdot \left(\frac{v_i}{v_i+c\cdot \mathbb{E}(V)}\right)^2 - \left(\sum_{i=1}^np_i\cdot \frac{v_i}{v_i+c\cdot \mathbb{E}(V)}\right)^2 $

Where, $p_i$ is the probability to obtain $v_i$

A solution for n=6 will be sufficient, although a general solution is also needed.


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Are the $v_i$ positive, integers, complex,...? – Joel Moreira Apr 10 '13 at 17:40
Hi, The values for $v_i$ are non negative integers, but you can treat them as real numbers if this allows for an answer. Thanks – user33118 Apr 15 '13 at 7:03
I'm pretty sure you do not expect anyone to write an exact formula for the root of the high degree polynomial that arises here, and the numeric maximization is routine, so what exactly are you looking for? – fedja May 27 '13 at 12:13

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