Proof

${Z}_{i}=\left({X}_{i}-{\mu }_{i}\right)/{\sigma }_{i},i=1,2$ are independent $N\left(0,1\right),$ and ${X}_{1}+{X}_{2}=\left({\mu }_{1}+{\mu }_{2}\right)+\left({\sigma }_{1}{Z}_{1}+{\sigma }_{2}{Z}_{2}\right).$ Transform $\left(\begin{array}{c}\hfill {Y}_{1}\hfill \\ \hfill {Y}_{2}\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill {\sigma }_{1}\hfill & \hfill {\sigma }_{2}\hfill \\ \hfill {\sigma }_{2}\hfill & \hfill -{\sigma }_{1}\hfill \end{array}\right)\cdot \left(\begin{array}{c}\hfill {Z}_{1}\hfill \\ \hfill {Z}_{2}\hfill \end{array}\right).$ Then $\left|det\left[\begin{array}{cc}\hfill {\sigma }_{1}\hfill & \hfill {\sigma }_{2}\hfill \\ \hfill {\sigma }_{2}\hfill & \hfill -{\sigma }_{1}\hfill \end{array}\right]\right|={\sigma }_{1}^{2}+{\sigma }_{2}^{2},\text{and}\left[\begin{array}{c}\hfill {Z}_{1}\hfill \\ \hfill {Z}_{2}\hfill \end{array}\right]=\frac{1}{{\sigma }_{1}^{2}+{\sigma }_{2}^{2}}\left[\begin{array}{cc}\hfill {\sigma }_{1}\hfill & \hfill {\sigma }_{2}\hfill \\ \hfill {\sigma }_{2}\hfill & \hfill -{\sigma }_{1}\hfill \end{array}\right]\cdot \left[\begin{array}{c}\hfill {Y}_{1}\hfill \\ \hfill {Y}_{2}\hfill \end{array}\right],$ and the joint pdf of Y1 = σ1Z1 + σ2Z2 and Y2 = σ2Z1σ1Z2 is

$\begin{array}{ll}\hfill {f}_{{Y}_{1},{Y}_{2}}& \left({y}_{1},{y}_{2}\right)\hfill \\ =\frac{1}{2\pi \left({\sigma }_{1}^{2}+{\sigma }_{2}^{}\right)}\hfill \end{array}$

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