Joint Density of N Dependent Uniformly Distributed Random Variables Could someone show me the formula with proof for the Joint Density and CDF for N uniformly distributed variables that are not necessarily independent?
Again, if certain forms of dependence are assumed, please point that out in the answer.
Thanks in advance ..
 A: In the case of two rvs' it is easy to show an example. Let $X$ be uniformly distributed over $[0,1]$. Then define $Y$ so that it be uniformly distributed over $[0,X]$
The dependence is obvious: $X$ limits $Y$'s possibilities.
One can create an $n$ dimensional model based on the example above.
However, I don't think that there is a general cook book formula for creating dependent rvs'.
A: There are many joint density functions of N randoms for which each of the randoms is uniformly distributed (on, for concreteness, the interval $[0,1)$), yet they variables are not independent.  The formula will depend on whi8ch jdf you choose.
Here is an extreme example:  The $N$ variables are labeled $X_k | k = 0 \ldots N-1$ and variable $X_0$ is chosen as a uniform variate on $(0,1)$. Then for all $k > 0$, 
$$
X_k = \left( X_0 + \frac{k}{N} \right) \pmod 1
$$
(where $a \pmod 1$ means the fractional part of $a$; $1.6 \pmod 1 = 0.6$). 
Here the jdf is 
$$
\text{jdf }(X_0 \ldots X_{N-1}) = \prod_{k=1}^{N-1} \left[ \delta\left(X_k - \frac{k}{N} -X_0 \right) + \delta\left(X_k - \frac{k}{N} - X_0 -1 \right)\right]
$$
