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 $Z=|u⋅v|^2$?

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.

Consider $u,v∈C^M$ 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 $Z=|u⋅v|^2$?

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 $Z=|u⋅v|^2$?