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For $0<𝑑≀1$, $𝑛$ a positive integer, and $π‘₯=(π‘₯_1,π‘₯_2,\ldots,π‘₯_𝑛)$ where $0≀π‘₯_𝑖≀1$ for $𝑖=1,2,…,𝑛$; what is the probability that $\max x <\frac{\Sigma x_𝑖}{𝑛 𝑑}$?

Monte Carlo experiments suggest the curves of the probabilities for fixed 𝑛 as functions of 𝑑 are smooth. The limit of these functions as 𝑛 increases without bound seems to be a step function. The main interest is to determine formulas for these functions, if they exist, as expressions in elementary functions. I'm curious to know the level of difficulty of this problem and its answer. I posted it in the hope someone would find it interesting. Drawing the $x_i$ from nonuniform distributions on the unit interval is an obvious generalization.

The problem arose in writing a computer code where such $π‘₯$ are desired to randomly choose initial values for some ODEs in a physical application. At present, I simply make a choice and test to see if it satisfies the desired inequality. If it doesn't, I choose again! A better code would use the answer to the posed question.

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Assuming that the coordinates are IID with CDF $F$, the CDF of their maximum is (wiki) $F^n$.

Meanwhile, the PDF of the sum of coordinates is given by the $n$-fold convolution of $F'$ (wiki). For the uniform distribution on coordinates, this yields the Irwin-Hall distribution.

Finally, the distribution of the ratio of the maximum and sum is a bit tricky, but doable (wiki).

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