Let $W = (1 - Z)X + ZY$ where $X \sim N(0, \sigma^2)$, $Y \sim N(\mu,\sigma^2)$ and $Z$ is Bernoulli with $P(Z = 0) = P(Z = 1) = 1/2$. $X,Y$ and $Z$ are independent. I try to find the distribution $p(Z|W)$. It seems $Z$ and $W$ are not independent. $p(Z\mid W) = \frac{p(Z,W)}{p(W)}$ does not really help. Any ideas to proceed?
Edit:
$$\begin{align} p_{Z|W}(z|w) &= \frac{p_{W|Z}(w|z)p_Z(z)}{p_W(w)} \\ p_W(w) &= p_{W|Z}(w|z=0)p_Z(z=0) + p_{W|Z}(w|z=1)p(z=1) \\ &= 0.5N(0, \sigma^2) + 0.5N(\mu, \sigma^2) \end{align}$$
Then: If $z = 1$, $$p_{Z|W}(z=1\mid w) = \frac{0.5N(\mu,\sigma^2)}{0.5N(0, \sigma^2) + 0.5N(\mu, \sigma^2)}$$ and if $z = 0$, $$\begin{align} p_{Z|W}(z = 0\mid w) &= \frac{0.5N(0,\sigma^2)}{0.5N(0, \sigma^2) + 0.5N(\mu, \sigma^2)} \\ &= \left.\frac{\frac{1}{\sqrt{2\pi\sigma^2}}\exp(\frac{-1}{2\sigma^2}(z-0)^2)}{0.5\frac{1}{\sqrt{2\pi\sigma^2}}\exp(\frac{-1}{2\sigma^2}(z-0)^2) + 0.5\frac{1}{\sqrt{2\pi\sigma^2}}\exp(\frac{-1}{2\sigma^2}(z-\mu)^2)} \right\vert_{z = 0} \end{align}$$