Pdf of weighted uniform random variables

[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!

 

[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.

 

[Stephen Tashi]

Is the question about the joint distribution of the y_i?

 

[PAHV]

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

 

[haruspex]

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

 

[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.