Question: Suppose X and Y are independent normal random variables. Let

\(\left\{ \begin{aligned} & Z=\alpha X+\beta Y \\ & W=\alpha X-\beta Y \\ \end{aligned} \right.\)

(where \(\alpha ,\beta \) are nonzero constants)

Given \({{\alpha }^{2}}\sigma _{x}^{2}={{\beta }^{2}}\sigma _{y}^{2}\) prove or disprove that Z and W are independent normal

random variables.