distribution of sum of two random variables having beta distribution.

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?

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