Consider $u,v∈S^{M-1}\subset \mathbb{C}^M$ to be two independent unit norm random vectors on the $M−1$ dimensional complex sphere $S^{M−1}$. In addition, $u$ follows an isotropic distribution, i.e., $u$ is uniformly distributed on the complex sphere $S^{M−1}$. What is the distribution of $Z=|u⋅v|^2$? This question has been asked before (Distribution of dot product of two unit random vectors), but I get a different result. I get that $Z$ follows Beta$(1,M−1)$ distribution by simulation.
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$\begingroup$ This question has been asked before, and answered, here: mathoverflow.net/questions/208937/… $\endgroup$– Yossi LonkeCommented Nov 20, 2018 at 14:48
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$\begingroup$ @Yossi Lonke，@Neil Hoffman，thank you very much firstly. I have read it before I asked this qustion, and the reason I asked agian is that I get a different distribution. I get that $Z$ follows Beta(1,M−1) distribution by simulation. I'm confused. $\endgroup$– QiangLiCommented Nov 21, 2018 at 1:19
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