Law of total variance with i.i.d. positive continuous r.v.s.

Solution 1:

$$\operatorname{Var}\left(Y\right) = \operatorname{E}\left[\operatorname{Var}\left(Y\mid X\right)\right] +\operatorname{Var}\left(\operatorname{E}\left[Y\mid X\right]\right)$$ If $Y$ is independent of $X$, then $$\operatorname{E}\left[\operatorname{Var}\left(Y\mid X\right)\right]=\operatorname{E}\left[\operatorname{Var}\left(Y\right)\right]$$ And $$\operatorname{Var}\left(\operatorname{E}\left[Y\mid X\right]\right)=\operatorname{Var}\left(\operatorname{E}\left[Y\right]\right)=0$$ Therefore $$\operatorname{Var}\left(Y\right) = \operatorname{E}\left[\operatorname{Var}\left(Y\right)\right]$$ Which makes sense because the expected value of a constant is equivalent to the constant. So yes, as others have already mentioned, your reasoning is indeed correct.