Will the conditional variance $V[Y|Z=z]$ depend on $z$ in this case?

Question

I start with two independent zero mean random variables $X$ and $Y$. They have variances $\sigma^2_X$ and $\sigma^2_Y$, respectively. Then I define a new random variable $$Z=0.6\,X + 0.4\, Y.$$ Is the conditional variance $V[Y|Z=z]$ constant or does $V[Y|Z=z]$ depend on $z$?

Answer 1

I think in general the variance does depend on $z$. Here is an example.

$X,Y\sim\mathcal{U}[-1,1]$. Then $Z=0.6X+0.4Y\in[-1,1]$. Given $z$, $Y$ is uniform on $[-1,\frac{10}{4}z+\frac{6}{4}]$ if $z\in[-1,-\frac{2}{10}]$ and on $[\frac{10}{4}z-\frac{6}{4}]$ if $z\in[-\frac{2}{10},1]$ (just draw $[-1,1]^{2}$). If $z\in\{-1,1\}$, $Y$ has zero variance, whereas if $z\in(-1,1)$, $Y$ has strictly positive variance.