Pdf of weighted uniform random variables

1. May 3, 2013

PAHV

Let x(1),...,x(N) all be independent uniformally distributed variables defined on (0,1), i.e. (x(1),...,x(N)) - U(0,1). Define the random variable y(i) = x(i)/(x(1)+...+x(N)) for all i=1,...,N. I’m looking for the pdf of the random variables y(1),…,y(N). Has anyone come across such random variables? If so, any references would be appreciated!

2. May 3, 2013

haruspex

I don't think it's too hard to work out. Start with deriving the CDF. P[Yi < y] = P[x(i)/(x(1)+...+x(N)) < y] = etc.

3. May 3, 2013

Stephen Tashi

Is the question about the joint distribution of the $y_i$?

4. May 7, 2013

PAHV

Yes, the question is about the joint distribution of the y(i). Any help on getting started with the pdf is highly appreciated!

5. May 10, 2013

haruspex

Try doing it for n=2. Even that is quite tricky. In general it looks very messy.

6. May 10, 2013

chiro

Hey PAHV and welcome to the forums.

If N is large enough, then you can approximate the sum by using the Central Limit Theorem (i.e. normal) approximation.

If not (or you are particular on having everything precise), then use the standard tools for calculating sum/ratio distributions.