Let $\left(X_i\right)_{i}^{n}$ be a family of independant random variables that follow a Bernoulli schemes. Let $U$ and $M$ be $$U= \begin{pmatrix} X_1\\ X_2\\ \vdots\\ X_n \end{pmatrix} \text{ and } M=U ^{t} \! U $$
I've found that $Rank\left(M\right)$ also follows a Bernoulli scheme of parameter $p'=1-\left(1-p\right)^n$ and that $Trace\left(M\right)$ follows a Bernoulli law $B\left(n,p\right)$. I would like to know, first if my results are right, and also if it was possible to excavate new results similar to this like the law of the determinant or even probability on its eigenvalue. ( It is always diagonalizable since it is symmetric with real entries )
Thanks