Calculating the expected value and variance of a sum of two different distributions I have the following expression:
$$
Z = G - N + E
$$
where
$$
G \sim \mathrm{Gamma}(k,\frac{\sigma^2}{k}) \\
N \sim \mathcal{N}(0,\dfrac{\alpha{}\sigma^2}{k^{2}}) \\
E \in \mathbb{R}_{+}
$$
This expression is obtained when the estimator of the variance for an IID normal distribution was used under the presence of a signal. More specifically,
$$
Z = \dfrac{\sigma^2}{2k}\sum_{n=0}^{k-1}U_{n}^{2} - \dfrac{\sqrt{\alpha\sigma^2}}{k}\sum_{n=0}^{k-1}U_{n}s[n] +\dfrac{\alpha}{2k}
$$
with $U \sim \mathcal{N}(0,1)$ and $\mathbf{s}^{T}\mathbf{s} = 1$.
I know that the expected value is a linear operator (i.e. $\mathbb{E}[X + Y] = \mathbb{E}[X] + \mathbb{E}[Y]$). The variance isn't a linear operator but it allows to decompose the expression further, to an extent.
Does this apply, however, in the case when the distributions are not similar? I have seen from some Googling that under the assumption of independence, the joint PDF was derived instead. I'm not sure if the two distributions are independent.

What needs analysis to be calcuated is $E(G N)$, but they can be deduced from the model to lead to the need to evaluate terms of the kind $c_{ij}E(U_i^2U_j)$ which is zero because all odd moments of the standard normal distribution is zero. hence
$$
Var(G+N) = E((G+N)^2) - ((E(G+N))^2 = E(G^2) + E(N^2) - ((E(G+N))^2
$$
