Let $N>0$ be the number of considered samples. We draw $x_1, \ldots, x_n$ from a uniform distribution over $[0;1]$. We compute $y_1, \ldots, y_{n-1}$ the differences of the sorted $(x_i)_i$.

I'd like to compute the expectancy of the max over the median for these values : $E(\max((y_i)_i) / \text{med}((y_i)_i)$

  • 2
    $\begingroup$ Luc Devroye has a chapter in Non-Uniform Random Variate Generation dedicated to Uniform and Exponential spacings (Chapter V). $\endgroup$
    – Xi'an
    Commented Jan 17, 2020 at 8:43
  • 2
    $\begingroup$ Adding $y_0=\min(x_i)$ and $y_n=1-\max(x_i)$ to the $y_i$'s, a convenient representation is for instance (Theorem 2.2, p.208):$$(Y_0,\ldots,Y_n)\sim(\epsilon_0,\ldots,\epsilon_n)\big/\sum_{j=0}^n \epsilon_j$$where the $\epsilon_j$'s are iid $\mathcal Exp(1)$. $\endgroup$
    – Xi'an
    Commented Jan 17, 2020 at 8:46


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