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 $\textbf{z}_{1}^{H}\textbf{z}_{1}, \ \textbf{z}_{2}^{H}\textbf{z}_{2}, \ \cdots, \ \textbf{z}_{P}^{H}\textbf{z}_{P} \sim \Gamma(M,2c)$.
$\textbf{EDIT: 30/12/2016}$: So far we now from What is the distribution of the ratio between independent Beta and Gamma random variables? the distribution of the ratio between independent Beta and Gamma random variables, $i.e.$,
\begin{equation}\label{eq:3} y_1 = \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_2 = \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.