I'm trying to figure out how to do deviation of the variance in the Moran model: https://en.wikipedia.org/wiki/Moran_process. The one thing that I don't understand is how do you get from $$E\left[\left(\frac{2Z}{N}\right)\left(1-\frac{Z}{N}\right)\right]+Var(Z)$$ to $$\left(\frac{2E[Z]}{N}\right)\left(1-\frac{E[Z]}{N}\right)+\left(1-\frac{2}{N^2}\right)Var(Z)$$ Where does $\left(1-\frac{2}{N^2}\right)$ come from?
I'd appreciate your help and thank you in advance!
You just need to use the relation $Var(Z) = E[Z^2] - (E[Z])^2$ as follows
\begin{aligned} E\left[\left(\frac{2Z}{N}\right)\left(1-\frac{Z}{N}\right)\right]+Var(Z)\quad & = E\left[\frac{2Z}{N}-\frac{2Z^2}{N^2}\right]+Var(Z)\\ &= \frac{2E[Z]}{N}-\frac{2E[Z^2]}{N^2}+Var(Z)\\ {\small\{\text{use $ E[Z^2] = Var(Z) + (E[Z])^2$ }\}}\qquad&= \frac{2E[Z]}{N}-\frac{2(Var(Z)+ (E[Z])^2}{N^2}+Var(Z)\\ & = \left(\frac{2E[Z]}{N}\right)\left(1-\frac{E[Z]}{N}\right)+\left(1-\frac{2}{N^2}\right)Var(Z) \end{aligned}