# Expectation of the ratio between Beta and Gamma random variables

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

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

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

I'd like to know the following expectation:

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

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, $i.e.$,

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

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

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

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.

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 $i \neq j$):

$$\mathbb{E} \left\lbrace y_2 \right\rbrace = \mathbb{E} \left\lbrace \frac{\textbf{z}_{i}^{H}\textbf{z}_{j}}{\left( \textbf{z}_{1}^{H}\textbf{z}_{1} + \textbf{z}_{2}^{H}\textbf{z}_{2} + \cdots + \textbf{z}_{P}^{H}\textbf{z}_{P} \right)^{2} } \right\rbrace = 0, \ \text{when} \ i \neq j.$$