Let $D$ be a Bernoulli random variable. When $D=1$, $Pr(D=1)=0.5$. When $D=0$, $Pr(D=0)=0.5$. Let $Z$ be a standard normally distributed random variable such that $Z \sim N(0,1)$.
We let $Y=D\times Z$. Is there any way to know the conditional distribution of $Y$ given $Z$? conditional expectation $\mathbf{E}(Y|Z)$ ? conditional variance $\mathbf{var}(Y|Z)$?
Intuitively, I would think the conditional distribution is still Bernoulli.
It looks to me that irrespective of the distribution of Z, any particular value of Z, say z leads to a random variable zY which is two valued $(\pm z)$, so $E(Y|Z)=\frac{z}{2}$ and $var(Y|Z)=\frac{z^2}{4}$.
The distribution function is obviously a little messy, Let $U=Y\times Z$, then $P(U)=0=0.5$ and for the other half, it has the same distribution as Z (multiplied by 0.5).