# How to find the distribution of the ratio of random vectors having two other known distributions?

I have the following problem. Consider a random vector distributed as a multivariate $t$-distribution, $\mathbf{v} \sim t_\nu(\mathbf{0}, \boldsymbol{\Omega})$. Consider further another random vector $w^{-1} \sim \mathcal{B}\textit{eta}\big(\frac{\nu}{2}, \frac{N}{2}\big)$ where $w = \big[1+ (1/\nu)(\mathbf{y} - \boldsymbol{\mu})^\top \boldsymbol{\Omega}^{-1} (\mathbf{y} - \boldsymbol{\mu})\big]$. I want to retrieve the distribution of the ratio $$\mathbf{G} = \frac{\mathbf{v}^{} \, \mathbf{v}^\top}{w}$$ in order to compute its moments, particularly, I'm intrested in its expected value, $\mathbb{E}[\mathbf{G}]$, because it will yields a covariance matrix. Any idea or suggestions for the calculation of the distribution of ratio of random vectors? Thanks a lot!

EDIT: I forgot to write that $\mathbf{v} = (\mathbf{y} - \boldsymbol{\mu})$ where $\mathbf{y} \sim t_\nu(\boldsymbol{\mu}, \boldsymbol{\Omega})$.

• I've finally found the solution! If anyone is interested, i can share without any problem! – Enzo D'Innocenzo Dec 28 '17 at 11:20