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?