Given
\begin{equation}\label{eq:definition\_of\_z}
\begin{split}
\textbf{Z} = \left[\begin{array}{cccc}
{z}\_{11} & {z}\_{12} & \cdots & {z}\_{1P} \\
{z}\_{21} & {z}\_{22} & \cdots & {z}\_{2P} \\
{z}\_{31} & {z}\_{32} & \cdots & {z}\_{3P} \\
\vdots & \vdots & \ddots & \vdots \\
{z}\_{M1} & {z}\_{M2} & \cdots & {z}\_{MP} \\ \end{array} \right] = \left[\begin{array}{cccc}
\textbf{z}\_{1} & \textbf{z}\_{2} & \cdots & \textbf{z}\_{P} \\ \end{array} \right]
\end{split}
\end{equation}


where $\textbf{z}\_{i}$ represents each of the columns of $\textbf{Z} \sim \mathcal{CN}(\textbf{0}\_{M \times P},\text{cM}\textbf{I}\_{P})$ and $z\_{ij} \sim \mathcal{CN}(0,c)$. 


Other necessary definition is given by


\begin{equation}\label{eq:definition\_of\_z\_z}
\textbf{Z}^{H} \textbf{Z} = \left[\begin{array}{c}
\textbf{z}\_{1}^{H} \\
\textbf{z}\_{2}^{H} \\
\vdots \\
\textbf{z}\_{P}^{H} \\ \end{array} \right] \left[\begin{array}{cccc}
\textbf{z}\_{1} & \textbf{z}\_{2} & \cdots & \textbf{z}\_{P} \\ \end{array} \right] = \left[\begin{array}{cccc}
\textbf{z}\_{1}^{H}\textbf{z}\_{1} & \textbf{z}\_{1}^{H} \textbf{z}\_{2} & \cdots & \textbf{z}\_{1}^{H} \textbf{z}\_{P} \\
\textbf{z}\_{2}^{H} \textbf{z}\_{1} & \textbf{z}\_{2}^{H} \textbf{z}\_{2} & \cdots & \textbf{z}\_{2}^{H} \textbf{z}\_{P} \\
\vdots & \vdots & \ddots & \vdots \\
\textbf{z}\_{P}^{H} \textbf{z}\_{1} & \textbf{z}\_{P}^{H} \textbf{z}\_{2} & \cdots & \textbf{z}\_{P}^{H} \textbf{z}\_{P} \\ \end{array} \right] 
\end{equation}


I'd like to know the following expectation:


\begin{equation}\label{eq:channel\_matrix}
 \mathbb{E} \left\lbrace \frac{ \textbf{Z}^{H} \textbf{Z} }{ \text{Tr} \left( \textbf{Z}^{H} \textbf{Z} \right)^{2} } \right\rbrace = \mathbb{E} \left\lbrace \frac{ \textbf{Z}^{H} \textbf{Z} }{ \left( \textbf{z}\_{1}^{H}\textbf{z}\_{1} + \textbf{z}\_{2}^{H}\textbf{z}\_{2} + \cdots + \textbf{z}\_{P}^{H}\textbf{z}\_{P} \right)^{2} } \right\rbrace
\end{equation}


where $\text{Tr}$ is the Trace operator. 


It is important to note that the elements of the main diagonal of $\textbf{Z}^{H} \textbf{Z}$, namely $\textbf{z}\_{1}^{H}\textbf{z}\_{1}, \ \textbf{z}\_{2}^{H}\textbf{z}\_{2}, \ \cdots, \ \textbf{z}\_{P}^{H}\textbf{z}\_{P} \sim \Gamma(M,2c)$.


So far I now from <https://stats.stackexchange.com/questions/253607/what-is-the-distribution-of-the-ratio-between-independent-beta-and-gamma-random> the distribution of the ratio between independent Beta and Gamma random variables, $i.e.$, 


\begin{equation}\label{eq:3}
y = \frac{\textbf{z}\_{i}^{H}\textbf{z}\_{j}}{\left( \textbf{z}\_{1}^{H}\textbf{z}\_{1} + \textbf{z}\_{2}^{H}\textbf{z}\_{2} + \cdots + \textbf{z}\_{P}^{H}\textbf{z}\_{P} \right)^{2} }, \ \text{when} \ i = j.
\end{equation}


What is still missing is the distribution of the following ratio:


\begin{equation}\label{eq:4}
y = \frac{\textbf{z}\_{i}^{H}\textbf{z}\_{j}}{\left( \textbf{z}\_{1}^{H}\textbf{z}\_{1} + \textbf{z}\_{2}^{H}\textbf{z}\_{2} + \cdots + \textbf{z}\_{P}^{H}\textbf{z}\_{P} \right)^{2} }, \ \text{when} \ i \neq j.
\end{equation}


Would the above function also have the same distribution as when $i = j$?


It's also worth mentioning that $\textbf{z}\_{1}^{H}\textbf{z}\_{1} + \textbf{z}\_{2}^{H}\textbf{z}\_{2} + \cdots + \textbf{z}\_{P}^{H}\textbf{z}\_{P}$ results in a scalar random variable.



Expectation of the ratio between Beta and Gamma random variables
For the following case I have found the following equation for its expectation


$$
 \mathbb{E} \left\lbrace y\_1 \right\rbrace = \mathbb{E} \left\lbrace \frac{\textbf{z}\_{i}^{H}\textbf{z}\_{j}}{\left( \textbf{z}\_{1}^{H}\textbf{z}\_{1} + \textbf{z}\_{2}^{H}\textbf{z}\_{2} + \cdots + \textbf{z}\_{P}^{H}\textbf{z}\_{P} \right)^{2} } \right\rbrace = \frac{2c}{P(MP-1)}, \ \text{when} \ i = j \ \text{and} \ MP \gt 1. $$


$\textbf{Proof}$:


Writing $y\_1$ as 


$$y\_1 = \frac{V}{U}$$


where $V = \frac{\textbf{z}\_{i}^{H}\textbf{z}\_{j}}{ \textbf{z}\_{1}^{H}\textbf{z}\_{1} + \textbf{z}\_{2}^{H}\textbf{z}\_{2} + \cdots + \textbf{z}\_{P}^{H}\textbf{z}\_{P} }$ and $U = \textbf{z}\_{1}^{H}\textbf{z}\_{1} + \textbf{z}\_{2}^{H}\textbf{z}\_{2} + \cdots + \textbf{z}\_{P}^{H}\textbf{z}\_{P}$ where $U \sim \Gamma(MP,2c) $ and $V \sim \beta(M,M(MP-1))$ are independent random variables.


Then taking into account that $V$ and $U$ are independent we have:


$$\mathbb{E} \left\lbrace y\_1 \right\rbrace = \mathbb{E} \left\lbrace \frac{V}{U} \right\rbrace = \mathbb{E} \left\lbrace V \right\rbrace \mathbb{E} \left\lbrace \frac{1}{U} \right\rbrace $$


Note that $\frac{1}{U}$ follows the inverse-gamma distribution and then if $U \sim \Gamma(MP,2c)$ thus $\frac{1}{U} \sim Inv-Gamma(MP,2c)$.


Next, the following expectations $\mathbb{E} \left\lbrace \frac{1}{U} \right\rbrace = \frac{2c}{MP-1}, MP \gt 1$ and $\mathbb{E} \left\lbrace V \right\rbrace = \frac{1}{P}$ give us the final result.


Finally, I'm supposing (but I'm not sure as I haven't figured out the distribution of $\textbf{z}\_{i}^{H}\textbf{z}\_{j}$ when $ i \neq j$):


$$
\mathbb{E} \left\lbrace y\_2 \right\rbrace = \mathbb{E} \left\lbrace \frac{\textbf{z}\_{i}^{H}\textbf{z}\_{j}}{\left( \textbf{z}\_{1}^{H}\textbf{z}\_{1} + \textbf{z}\_{2}^{H}\textbf{z}\_{2} + \cdots + \textbf{z}\_{P}^{H}\textbf{z}\_{P} \right)^{2} } \right\rbrace = 0, \ \text{when} \ i \neq j.
$$


