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If we are given a fixed integer $N > 0$ of choices we can pick out of a pool of $k$ values $c_0, \cdots, c_k$ (with repetitions allowed and $c_i > 0 \forall i$) and we want to maximize the expectation of:

$$f(x) = \sum_{i = 0}^k n_i c_i X(i)$$

where the $c_i$ are from above and $n_i$ is how many $c_i$ we picked (so obviously $\sum_i n_i = N$) and $X$ is a bernoulli random variable which takes the value $1$ with probability $p(i)$ ($p$ a given decreasing function of $i$)?

What if we allow the $c_i$ to vary across a continuous range $[0,k]$ and make $p(i)$ continuous as well? Does this simplify the optimization?

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What is $x$? If $X$ is Bernoulli, what is $X(i)$? If you want to maximize $\sum\limits_in_ic_ip(i)$, why not taking $N$ times $c_i$ such that $c_ip(i)$ is maximal? –  Did Jul 8 '12 at 10:13
Abandon the question, did you? –  Did Jul 14 '12 at 14:22
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