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Let $U_1, \ldots, U_n$ be $n$ i.i.d discrete uniform random variables on (0,1) and their order statistics be $U_{(1)}, \ldots, U_{(n)}$.

Define $D_i=U_{(i)}-U_{(i-1)}$ for $i=1, \ldots, n$ with $U_0=0$.

I am trying to figure out the joint distribution of $U_i$'s and their marginal distribution and possibly their first few moments. Can anyone give some hint on this. Also can you please recommend a book on order statistics?

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  • $\begingroup$ By "discrete uniform random variables on (0,1)" do you mean Bernoulli random variables? In that case the joint distribution of the $U_i$ is almost trivial. Or do you have something else in mind? $\endgroup$
    – Jonathan Christensen
    Commented Jan 18, 2013 at 5:03
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    $\begingroup$ If you meant continuous uniform on $(0,1)$ then $(D_1, \ldots, D_n)$ has a Dirichlet distribution. $\endgroup$
    – Stéphane Laurent
    Commented Jan 18, 2013 at 8:10

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There are many papers addressing such questions.

A good starting place is probably:

Pyke R. (1965), Spacings
Journal of the Royal Statistical Society. Series B (Methodological)
Vol. 27, No. 3 (1965), pp. 395-449

(It has a lot on the continuous case. Many papers refer to this paper, including some that do more with the discrete case.)

You should be able to read it online:

http://www.jstor.org/discover/10.2307/2345793

(for me it says 'read online free' without me being logged into any institutional access)

For continuous uniform distributions, the answers are easy. For discrete distributions, accurate answers are much harder, though if the discrete uniform takes many different values, the continuous calculation can sometimes be a reasonable approximation.

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