I'm asking for someone to check my work on this problem.
Wikipedia gives the formula $$ \mathsf{Var} Y=\mathsf E(\mathsf{Var}(Y|X_1,X_2))+\mathsf E(\operatorname {Var}(\mathsf E (Y|X_1,X_2)|X_1))+\mathsf{Var}(\mathsf E (Y|X_1)). $$ Now suppose $Y\sim\mathcal N(X_1,X_1+X_2)$ and $X_i\sim\operatorname{Gamma}(\alpha_i,\alpha_i/\sigma_i^2)$ with $X_1$ and $X_2%$ independent and paramterized in terms of shape $\alpha_i$ and rate $\alpha_i/\sigma_i^2$. I want to make sure I'm applying the above formula correctly to this problem.
Here is what I have: $$ \begin{align} \mathsf{Var} Y &=\mathsf E(X_1+X_2)+\mathsf E(\operatorname {Var}(X_1|X_1))+\mathsf{Var}(\mathsf E(\mathsf E (Y|X_1,X_2)|X_1))\\ &=\sigma_1^2+\sigma_2^2+\mathsf E(0)+\mathsf{Var}(\mathsf E(X_1|X_1))\\ &=\sigma_1^2+\sigma_2^2+0+\mathsf{Var}X_1\\ &=\sigma_1^2+\sigma_2^2+\sigma_1^4/\alpha_1. \end{align} $$
Is this correct?