Conditional Variance of a linear combination

Question

Let X,Y be two R.V. Is the following correct?

$Var[aX+bY|Y]=Var[aX|Y] +Var[bY|Y]+2Cov[aX,bY|Y]$?

Where $Var[bY|Y]=0$.

Answer 1

If $X$ and $Y$ are dependent, then you cannot make such a statement. All you can write is $$\operatorname{Var}[a X + b Y \mid Y] = a^2 \operatorname{Var}[X \mid Y].$$ There is no further simplification possible because the dependence of $X$ upon $Y$ is not specified.

If $X$ and $Y$ are independent, then $$\operatorname{Var}[a X + b Y \mid Y] = a^2 \operatorname{Var}[X].$$