Pdf of weighted uniform random variables

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Discussion Overview

The discussion centers around the probability density function (pdf) of weighted uniform random variables defined as y(i) = x(i)/(x(1)+...+x(N)), where x(1),...,x(N) are independent uniformly distributed variables on (0,1). Participants explore the derivation of the pdf and the joint distribution of these random variables.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant seeks references for the pdf of the random variables y(1),…,y(N).
  • Another suggests starting with the cumulative distribution function (CDF) to derive the pdf.
  • A question is raised about whether the inquiry pertains to the joint distribution of the y(i).
  • Clarification is provided that the focus is indeed on the joint distribution of the y(i).
  • One participant notes that calculating the pdf for N=2 is quite challenging and suggests that the general case appears complex.
  • Another participant mentions that for large N, the sum can be approximated using the Central Limit Theorem, while also indicating that standard tools for calculating sum/ratio distributions may be necessary for precise results.

Areas of Agreement / Disagreement

Participants express varying levels of confidence regarding the complexity of deriving the pdf, with some suggesting approximations and others emphasizing the challenges involved. No consensus is reached on a specific method or solution.

Contextual Notes

Participants acknowledge the potential complexity of the problem and the need for precise calculations, particularly when N is small. The discussion reflects uncertainty about the best approach to derive the pdf.

Who May Find This Useful

Readers interested in probability theory, particularly those studying the properties of random variables and their distributions, may find this discussion relevant.

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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