# Distribution of dot product of two unit complex random vectors [duplicate]

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.

• This question has been asked before, and answered, here: mathoverflow.net/questions/208937/… – Yossi Lonke Nov 20 '18 at 14:48
• @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. – QiangLi Nov 21 '18 at 1:19