I have this proceses $\sum_{n=0}^{N/2-1} z(n)$ where \begin{equation}\label{eq:z(n)} 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}
$\mathtt{p}$ is a random variable that depends on $z(n)$. For instance to find $P\text(\mathtt{p}=0)$:
$\tilde{s} _1=\dot{s}+z_1(n)$ and $\tilde{s} _2=\dot{s}+z_2(n)$
\begin{align}P_c( 2\tilde{s} _2<\tilde{s} _1\cap \tilde{s} _1=\mathtt{s}\vert \dot{s}>0) &=P( 2\tilde{s} _2<\tilde{s} _1)P(\tilde{s} _1=\mathtt{s}\vert \dot{s}>0) \\ \nonumber &= f_{\tilde{s}_1} (\mathtt{s}; s_1,\sigma_1\vert \dot{s}>0) \Phi (\mathtt{s}/{2\sigma_2)}, \end{align}
\begin{align*} P\text(\mathtt{p}=0) & = {\int_{-\infty}^{\infty}} P_c( 2\tilde{s} _2<\tilde{s} _1\cap \tilde{s} _1=\mathtt{s}\vert \dot{s}>0) \, d\mathtt{s}. \end{align*}
I need to find the variance of the process. Is there any ideas where I should start from?
I thought of doing the following. Let $\sigma_1^2$ be the variance of $z_1(n)$, and $\sigma_2^2$ be the variance of $-(z_1(n)+z_2(n))/\sqrt{2}$. Then the variance of $z(n)$ is $$ P\text(\mathtt{p}=0) \sigma_1^2 + \mathbb{P}(p=1) \times \sigma_2^2 $$ The challenge in my problem is that to find $\mathtt{p}=1$ is found by $P_c( 2\tilde{s} _2>\tilde{s} _1\cap \tilde{s} _1=\mathtt{s}\vert \dot{s}>0)$ so $P_c$ can be easily found, but it is not mapped to $\tilde{s} _1=\mathtt{s}$ instead its mapped to $(\tilde{s} _1+\tilde{s} _1)/\sqrt{2}$. I know the pdf of $(\tilde{s} _1+\tilde{s} _1)/\sqrt{2}$, if that helps