I want to calculate the expectation and variance of this process, can I assume that this Binomial and find $E(x)=p(\mathtt{p}=0)E(z_1(n))+(1-p(\mathtt{p}=0))E(-(z_1(n)+z_2(n))/\sqrt{2})$ and apply the same for variance as in Binomial variables?
note that $z_1(n)$ and $z_2(n)$ are i.i.d normal distributed random variables with zero mean and $\sigma=1$
$x={\sum_{n=0}^{N/2-1}} z(n),$ \begin{equation} z(n) = \begin{cases} z_1(n), & \textrm{if} \, \mathtt{p}=0 ;\\ -(z_1(n)+z_2(n))/\sqrt{2} , & \textrm{otherwise}. \end{cases} \end{equation}
\begin{equation}\label{eq:detectedpolarmetric} \mathtt{p}= \begin{cases} 0, & \textrm{if} \,\, [z_1(n)-z_2(n) ] > z_2(n);\\ 1, & \,\, \textrm{otherwise}. \end{cases} \end{equation}