How to show $\mathrm{Var}(\mathrm{E}(\sum_{i=1}^y X_i|Y)) = \mathrm{E}(X)^2\mathrm{Var}(Y)$

Question

Is this the correct approach? I'm not sure about the last step, is $\mathrm{Var}(y) = \mathrm{Var}(Y)$? (All $X_i$ have same distribution, and $Y$ is a random variable.

\begin{align*}\mathrm{Var}\left(\mathrm{E}\left(\sum_{i=1}^Y X_i\mid Y\right)\right) &= \mathrm{Var}\left(\mathrm{E}\left(\sum_{i=1}^Y X_i\right)\right)\\&= \mathrm{Var}\left(Y\mathrm{E}\left(X\right)\right) \\&=\mathrm{E}(X)^2\mathrm{Var}(Y). \end{align*}

Answer 1

Just add a little bit more details to make the proof more clear

$Var(E(\sum^{y}_{i=1}X_i|Y))=Var(E(\sum^{Y}_{i=1}X_i))=Var(\sum^{Y}_{i=1}EX_i)$

Let $EX_i=\mu$, where $\mu$ is a constant,

$Var(\sum^{Y}_{i=1}EX_i)=Var(\sum^{Y}_{i=1}\mu)=\mu^2Var(Y)=E(X)^2Var(Y)$

By the way, of course $EX_i$ should exist.