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Consider random variables $X_1, X_2, \ldots, X_n$ that are IID following a uniform distribution $\text{Unif}(0, 1)$.

What is the probability that the following event occurs:

$$ |X_1 - X_2| < \frac{1}{c}, \quad |X_1 - X_3| < \frac{1}{c}, \quad \ldots, \quad |X_{n-1} - X_n| < \frac{1}{c}? $$

After trying to figure this out I have come to the conclusion that I need to calculate the ration of a volume in $\mathbb{R}^n$ to $1$, but I can not find this area for general n. For $n=2$ this volume appears to be $1 - (1 - \frac{1}{c})^2$. For the case of $n=3$ I have worked out that the probability should be $\frac{1}{6c^3}*(9c-4)$ but my simulation seems to disagree.

Any help on the general case, or even on the case of $n=3$ would be appreciated. Thanks in advance.

Todor Slavov
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    The second one should be $|X_2 - X_3|$, right? – leonbloy Aug 09 '24 at 13:18
  • This should be relevant: https://math.stackexchange.com/questions/4740702/distribution-of-the-largest-gap-between-uniform-random-variables – leonbloy Aug 09 '24 at 13:26
  • What if you define $Y=\min[X_1, ..., X_n]$ and you condition on $Y$ being a value $y \in [0,1]$? Note the cases $y<1-1/c$ and $y\geq 1-1/c$ can be treated separately. – Michael Aug 09 '24 at 17:12
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    Check this out: https://math.stackexchange.com/a/2001026/321264. Your probability is \begin{align} P\left(\bigcap_{i=1}^{n-1} \left{|X_i-X_{i+1}|< \frac1c\right}\right) &=1-P\left(\bigcup_{i=1}^{n-1} \left{|X_i-X_{i+1}|> \frac1c\right}\right) \&=1-P\left(\min\limits_{1\le i\le n-1}|X_i-X_{i+1}| > \frac1c \right). \end{align} – StubbornAtom Aug 12 '24 at 10:29

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