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i have two random variables X,Y where

$X=\frac{|U^TV|^2}{||U||^2||V||^2}$

$Y=\frac{|U^TW|^2}{||U||^2||W||^2}$

U,V,W $\in R^{2M}$ and all are independent , zero mean gaussian. Also, $||V||^2=||W||^2$ also they are orthogonal to each other and X,Y follows beta distribution~B(0.5,M-0.5). Experimentally i've found that X and Y are not independent and X+Y follows ~B(1,M-1) distribution. for my current work, it will be helpful if i can prove this.

any suggestions about the pdf of Z=X+Y?

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  • $\begingroup$ Please read the description of the tag (distribution-theory) before using it in cases where it is irrelevant. Oh, and... where is your input? $\endgroup$ – Did Feb 1 '18 at 9:26
  • $\begingroup$ sorry, i didn't get that, what do you mean by input? we know the distribution of X,Y $\endgroup$ – ishmam zabir Feb 1 '18 at 9:40
  • $\begingroup$ Since we know the PDFs of X and Y independent from those of U,V,W , why does their PDF matter here to be mentioned? $\endgroup$ – Mostafa Ayaz Feb 1 '18 at 10:32
  • $\begingroup$ pdf of U,V,W doesn't matter. $\endgroup$ – ishmam zabir Feb 1 '18 at 10:50
  • $\begingroup$ "Input" = Almost anything you would contribute which would not be simply uploading your homework to the site. $\endgroup$ – Did Feb 1 '18 at 14:14

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