Pdf of weighted uniform random variables

AI Thread Summary
The discussion focuses on finding the probability density function (pdf) of random variables defined as y(i) = x(i)/(x(1)+...+x(N)), where x(i) are independent uniform random variables on (0,1). Participants emphasize the importance of deriving the cumulative distribution function (CDF) to approach the problem. The joint distribution of the y(i) is also a key concern, with suggestions to start with the case of N=2 for simplicity. For larger N, the Central Limit Theorem can be applied for approximation, while precise calculations require standard methods for sum and ratio distributions. Overall, the complexity of deriving the pdf is acknowledged, with requests for references and guidance.
PAHV
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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!
 
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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.
 
Is the question about the joint distribution of the y_i?
 
Yes, the question is about the joint distribution of the y(i). Any help on getting started with the pdf is highly appreciated!
 
Try doing it for n=2. Even that is quite tricky. In general it looks very messy.
 
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.
 
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