Expected value of nearest distance between random uniform variables I have $X_1,...,X_N$ independent random variables, with $X_i \sim U([0,1])$ $\forall i$.
I'm trying to answer the following question, informally speaking: suppose I am in $X_i$, what is the mininum distance I should travel in order to find the nearest $X_j$, for some $j$ ?
I formalised in the following way: find $\mathbb{E}[D_i]$, $D_i=\min_{j\neq i}\left \{ | X_i - X_j | \right \}$.
Now, doing some computations with $i\neq j$ we have that $Y_{ij}=| X_i - X_j |$ is continuous with probability density $f_{Y_{ij}}(y)=2(1-y)\mathbb{I}_{[0,1]}(y)$.
To compute $\mathbb{E}[D_i]$ one could compute $\mathbb{P}(D_i\geq d)=\mathbb{P}(\bigcap_{j\neq i}\{Y_{ij}\geq d\})$.
And here come the problems, since the variables $Y_{ij}$ (for a chosen $i$ and $j\neq i $) should not be independent (?).
So, anybody could help me from here? Maybe something clever with characteristic functions? Furthermore, is the problem correctly formalised?
Thank you all in advance!
 A: An outline that may possibly help is the following.


*

*For $i=2,\ldots,N$,  let $D_{i,j} = \underset{j<i}{\min}\{|X_i-X_j|\}$ since $(\forall i\neq j),\,|X_i - X_j| = |X_j - X_i|$. It follows that, $D_j = D_{(j+1,\,j)}$ where $D_{(k,j)}$ is the $k$-th order statistic of the set $\{D_{j+1,\,j},D_{j+2,\,j},\ldots,D_{N,\,j}\}$. 

*Since $\{D_{i,\,2}\}_{i=3}^{N}, \{D_{i,\,3}\}_{i=4}^{N},\ldots,\{D_{i,\,N}\}_{i=N-1}^{N}$ are independent sets, it follows that $\mathcal{D}_{N-1} = \{D_{1}, D_{2},\ldots, D_{N-1}\}$ form a collection of independent random variables.

*If we could show that the elements of $\mathcal{D}_{N-1}$ are all identically distributed say by $f_{\mathcal{D}_{N-1}}(t)$, then it immediately follows that $\mathbb{E}[D_i] = \mathbb{E}[D_1]$ with expectation taken with respect to $f_{\mathcal{D}_{N-1}}(t)$.

*If the elements of $\mathcal{D}_{N-1}$ are not identically distributed, then we obtain the joint $f_{\mathcal{D}_{N-1}}(d_1,\ldots,d_{N-1}) = \prod_{i=1}^{N-1}f_{D_i}(d_i)$. The value of $\mathbb{E}[D_i]$ is the corresponding marginal expectation with respect to $f_{D_i}(d_i)$.

