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We have a random variable $X\sim Pois(\lambda)$ with $\lambda$ unknown, and given any random sample $S=\{s_1, \ldots, s_{|S|}\}$ generated from $X$, we define $$\text{numsum}(S)=\sum_{i=1}^{|S|}s_i$$.

Our problem is then,

Given a natural number $L$ and a real number $p\in[0, 100]$, we have to find the biggest natural number $M$ such that if I generate random numbers from $X$ and accumulate them in a set $S_1$ until the set has a size $M$, and then I construct a new set $S_2$ and start accumulating numbers again until $S_2$ has a size $M$, and I repeat this process indefinitely, generating an infinite amount of sets of size $M$, $\text{numsum}(S_i)\leq L$ holds for $p\%$ of the sets (or better).

To estimate $M$, assume we have an initial random sample $Y=\{y_1, \ldots, y_{|Y|}\}$ available where $|Y|$ is under your control (you can get an initial random sample has large as you need, although the shorter the better).

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  • $\begingroup$ Isn't this just a restatement of your previous question at stats.stackexchange.com/questions/563926/…?? $\endgroup$
    – whuber
    Feb 13 at 21:31
  • $\begingroup$ @whuber Yes but I thought I had deleted the old one already (and I just did) and that is why I asked again with a much less ambiguous specification of the problem in my opnion. $\endgroup$ Feb 13 at 21:43
  • $\begingroup$ @whuber besides in my previous question I said "lowest M" which is exactly the opposite that I really want to ask (biggest M), because if I ask for the lowest M, with M = 0 the bound holds 100% of the time. $\endgroup$ Feb 13 at 21:45
  • $\begingroup$ If $p$ can be as large as $100$, then $100p\%$ can be as large as $10000\%$ $\endgroup$
    – Glen_b
    Feb 14 at 0:13
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    $\begingroup$ @Glen_b, that is what happens when one rewrite the questions dozens of tomes. Fixed. $\endgroup$ Feb 14 at 0:15

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