1

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 ..

texmex
  • 800

2 Answers2

1

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'.

zoli
  • 20,452
  • Please note, N could be 2, but is more likely to be more than 2. – texmex May 06 '15 at 14:04
  • Thanks Zoli, Could you please elaborate on how could one create an n dimensional model based on the above example and give complete details of the derivation or point to a source? What are the other forms of commonly used dependence structures? – texmex May 06 '15 at 14:05
  • Yes, @ravi.all.over. I modified my answer accordingly. ($X_1$ is a bound for $X_2$, $X_2$ is a bound for $X_3$, and so on. – zoli May 06 '15 at 14:06
1

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 $$ \text{jdf }(X_0 \ldots X_{N-1}) = \prod_{k=1}^{N-1} \left[ \delta\left(X_k - \frac{k}{N} -X_0 \right) + \delta\left(X_k - \frac{k}{N} - X_0 -1 \right)\right] $$

Mark Fischler
  • 41,743
  • Thanks Mark. What is the delta here? – texmex May 06 '15 at 14:14
  • The delta function enforces that its argument be zero; thus for each $k$, $X_k - \frac{k}{N} = X_0$. You find delta functions frequently when describing probability density functions relating to a discrete distribution. – Mark Fischler May 10 '15 at 04:00
  • Thanks Mark, Could you please address this other question? I am not sure if I should post this question here … But they are related … http://math.stackexchange.com/questions/1272351/n-affiliated-random-variables-conditional-density-and-distribution-of-first-ord – texmex May 10 '15 at 04:06