Theory and Methods of Statistics

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 = σ₂Z₁ − σ₁Z₂ is

$\begin{array}{ll}\hfill f_{Y_1,Y_2}& \left(y_1,y_2\right)\hfill \\ =\frac{1}{2\pi \left({\sigma }_1^2+{\sigma }_2^{}\right)}\hfill \end{array}$