Expectation of the ratio between Beta and Gamma random variables 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.
 A: 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$):
$$
\mathbb{E} \left\lbrace y_2 \right\rbrace = \mathbb{E} \left\lbrace \frac{\textbf{z}_{i}^{H}\textbf{z}_{j}}{\left(   \textbf{z}_{1}^{H}\textbf{z}_{1} + \textbf{z}_{2}^{H}\textbf{z}_{2} + \cdots + \textbf{z}_{P}^{H}\textbf{z}_{P}     \right)^{2} } \right\rbrace = 0, \ \text{when} \  i \neq j.
$$
