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Math Help - Distributing arbitrary values over a given range and target mean

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    Distributing arbitrary values over a given range and target mean

    Hey folks,

    Forgive my lack of math nomenclature and probably ham-fisted notation/explanation; I'm very very rusty in this department. First post here, and I'm liking the LaTeX.

    Larger problem:

    I need possible approaches to re-distribute a set of positive real numbers \mathcal{S} such that 0\leq s <\infty, s \in \mathcal{S}.

    I need the re-distributed set \mathcal{T} to follow:

    • the same ordering
    • to be in the interval min\leq t \leq 1 for a given value min where 0\leq min < 1
    • the mean of \mathcal{T} is a given value mean where 0\leq mean < 1

    Sub-problem

    In fact, I would also be happy if there's an approach which satisfies the case where min=0. My solution in this case was to divide all the elements of \mathcal{S} by the max value giving \mathcal{S}_\prime (to get between 0 and 1) and to apply some power x such that:

    \displaystyle\sum_{s_{\prime}\in\mathcal{S}_\prime  } s_{\prime}^{x} = mean * |\mathcal{S}_\prime|

    But, I have no idea how to solve for x given \mathcal{S}_\prime and mean.

    Any help for either suggesting an approach to creating \mathcal{T} from \mathcal{S} or solving for x would leave me fawning in gratitude.
    Last edited by ihavethedumb; January 10th 2010 at 11:05 AM. Reason: mean * |S| not just mean
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