Re: ask for a proof

> Hi,
>
> I am looking for a proof of the following problem.
>
> x1,x2,... xn are i.i.d. of exponential distribution.
> Define
> yi = xi/(x1+x2+...+xn), i = 1,2,...,n.
> Hence we know that
> the n-vector y=[y1,y2,..., yn] falls in the set
> A={[y1,y2,..., yn] :yi\le 0, \sum_i yi =1}
> I guess that y is distributed uniformly in set A. But
> how to prove it?
>
> Thank you!
>


What you have is a special case of the Dirichlet distribution which can be constructed when each of the x(i) is a gamma.

For exponentials, it reduces to y(1), ..., y(n-1) being uniform within 0 </= Sum(y(i)) </= 1. (The sum is over i = 1,, ..., n-1.)

Since y(n) = 1 - Sum(y(i)), the y(i) are uniform on the (n-1)-dimensional set A.

As far as the proof is concerned, start off with the joint density of the n exponentials x(i). Transfer to y(1), ..., y(n-1), x(n) using the Jacobian. Then integrate out x(n) and you should end up with the (n-1)-dimensional uniform distribution for y(1), ..., y(n-1).

Jack
.

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