What is the conditional $\operatorname{Var}(XY|Y)$ given that $X$ and $Y$ are independent?

Question

What is the conditional $\operatorname{Var}(XY|Y)$ given $X$ and $Y$ are independent? Is it:

$$\operatorname{Var}(XY|Y)= Y^2\operatorname{Var}(X|Y) = Y^2\operatorname{Var}(X)?$$

Answer 1

Your reasoning flow shows that you had a good understanding on conditioning. Formally, you can derive it from the definition$^\dagger$ of conditional variance and basic properties of conditional expectation: \begin{align} & \operatorname{Var}(XY|Y) \\ =& E[X^2Y^2 | Y] - (E[XY|Y])^2 \tag{definition}\\ =& Y^2E[X^2|Y] - (YE[X|Y])^2 \tag{pulling out known factors} \\ =& Y^2E[X^2] - Y^2(E[X])^2 \tag{independence}\\ =& Y^2\operatorname{Var}(X). \end{align}

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$^\dagger$ For two random variables $\xi$ and $\eta$ such that $E[\xi^2] < \infty$, the conditional variance of $\xi$ given $\eta$ is \begin{align} \operatorname{Var}(\xi|\eta) = E[(\xi - E[\xi|\eta])^2|\eta] = E[\xi^2|\eta] - (E[\xi|\eta])^2. \end{align}