Distribution of $s^*w$ if $s$ and $w$ are i.i.d. isotropic unit norm complex vectors - Cross Validated most recent 30 from stats.stackexchange.com 2019-08-23T18:11:45Z https://stats.stackexchange.com/feeds/question/182436 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://stats.stackexchange.com/q/182436 1 Distribution of $s^*w$ if $s$ and $w$ are i.i.d. isotropic unit norm complex vectors tam https://stats.stackexchange.com/users/48775 2015-11-18T18:46:05Z 2015-11-30T14:33:47Z <p>Let $w \in \mathbb C^M$ be a unit norm complex vector. Also, let $s \in \mathbb C^M$ be a unit norm complex vector independent of $w$. We assume that $s$ and $w$ are i.i.d. isotropic vectors. </p> <p>I am looking for the distribution of $s^* w$, where $s^*$ denotes the conjugate transpose of $s$. I also want to deduce the distribution of $|s^* w|$ and $|s^* w|^2$.</p> <p>I saw the claim that $|s^* w|^2\sim\mathsf{Beta}(1,M-1)$ but without a proof.</p> <p>There is a similar thread <a href="https://stats.stackexchange.com/questions/85916">Distribution of a scalar product of two random unit vectors in $\mathbb{R}^D$</a> that shows that for real vectors scalar product $t =s^\top w$ is distributed such that $$(t+1)/2 \sim \mathsf{Beta}\big((D−1)/2,(D−1)/2\big),$$ but I am not sure how to generalize that to complex vectors.</p>