I am working on research project related to wireless communication wherein there is extensive use of concept of random variables.
I have the following equation: $X = h_0h_{1_{n}}$ ---(1)
where, $h_0$ is Nakagami-$m$ random variable with mean power $\Omega_0$ and $h_{1_{n}}$ is Gaussian noise with mean power $\sigma^2$.
My query is what will be variance of $X$ of equation (1).
Any help in this regard will be highly appreciated.
I try to reformulate the variables at play to make sure I got it right:
$$ h_0 \sim \text{Nak}(m, \Omega_0) \\ h_1 \sim \mathcal{N}(0,\sigma^2) $$
Assuming the variables $h_0$ and $h_1$ are independent (and hence we can factor the expectation of the product) we have:
$$ \mathbb{E}(X)=\mathbb{E}(h_0 h_1) = \mathbb{E}(h_0)\mathbb{E}(h_1)=0 \\ \mathbb{E}(X^2)=\mathbb{E}(h_0^2 h_1^2) = \mathbb{E}(h_0^2)\mathbb{E}(h_1^2)=\Omega_0 \sigma^2 \\ \text{Var}(X)=\mathbb{E}(X^2)-[\mathbb{E}(X)]^2=\Omega_0 \sigma^2 $$
where for the second moment we used the fact that for $Z\sim \text{Nak}(m, \Omega)$ we have $\mathbb{E}(X^2)=\Omega$