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Let $\xi_1,\xi_2 \sim \mathcal{N}(0,1)$. Then $X = \alpha + \xi_1 \sqrt{\beta}$ and $Y = \mu X + \xi_2 X$ $ = \mu \alpha + \mu \xi_1 \sqrt{\beta} + \xi_2 \alpha + \xi_1 \xi_2 \sqrt{\beta}$. Hence $E(Y) = \mu \alpha$, $Var(Y) = \mu^2 \beta + \alpha^2 + \beta$.