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There are $N$ people. Each person has a paper with a numbered score $\in [1, T]$ where $T \geq N$.

A $Q$ percentage of people have papers with the same score, with $1 - Q$ percentage of people having papers with uniformly distributed unique scores.

Ask $k < N$ people for their papers.

We would then normalize the scores of all $k$ papers we receive with respect to the sum of all $k$ papers scores.

More explicitly, if $X$ is a column vector whose $i$'th entry is paper $i$'s score, where $i \in [1, k]$:

$$X' = \frac{X}{\sum_{i=1}^{k}{X_i}}$$

... represents the normalized $k$ papers scores.

Denote $W$ to be a column vector representative of normalized weights $\in [0, 1]$.

$$ W = \frac{X - \min_{X_i \in X}{X_i}}{\max_{X_i \in X}{X_i} - \min_{X_i \in X}{X_i}} $$

Now, the question is:

For any uniformly randomly sampled $k$ papers, what is the probability that:

$$\exists \text{score} \in X'W > \frac{2 \alpha}{\max{(2, C)}} $$

... where $C$ is the number of $k$ papers received with unique scores, and $\alpha \in [0, 1]$?

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  • $\begingroup$ Could you define $C$, and what you mean by $\exists\text{score}\in X'\cdot > ...$? Also, I may be wrong, but doesn't this depend on the distribution of the scores on $[1,T]?$ Can we make the additional assumption that $X_i \sim \text{Unif}([0,T])$, which is different from a subset of size $k$ being chosen with uniform probability $1/\binom{N}{k}$? $\endgroup$ – snar Dec 4 '19 at 5:07
  • $\begingroup$ @snar $C$ is defined at the bottom being the number of $k$ papers received with unique scores in a single sampling. Also, clarified in the question that unique scores denoted by $X_i$ are uniformly distributed! $\endgroup$ – Dranithix Dec 4 '19 at 5:26
  • $\begingroup$ OK, but what does $\exists \text{score} \in X' \cdot W > ...$ mean? Is $X'\cdot W$ an inner product? What does membership to a scalar mean? $\endgroup$ – snar Dec 4 '19 at 14:36
  • $\begingroup$ @snar just scalar multiplication s.t. $X'W$ - I'll clarify that in the question. $\endgroup$ – Dranithix Dec 5 '19 at 0:04

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