Find $\theta$ such that $W = X \cosθ +Y \sinθ \text{ and } Z = X \cosθ −Y \sinθ$ are independent. It is given that X and Y be jointly normal each with mean $0$ and variance $1$.
I have shown that $$W \sim N(0, 1)$$ $$Z \sim N(0, 1)$$
Now, how can I proceed from here?
Since $X$ and $Y$ are bivariate normal distribution.
Theorm: Let $(X_1,X_2,...,X_n)$ be an n-dimensional RV with a normal distribution. Let $Y_1,Y_2,...,Y_k, k\leq n,$ be linear functions of $X_j (j = 1,2,...,n)$. Then $(Y1,Y2,...,Yk)$ also has a multivariate normal distribution.
$W$ and $Z$ are also the linear combinations of X and Y. Hence we can use the above theorem.
$$\mathbb{E}[WZ] = 0$$ $$\mathbb{E}[X^2\cos^2\theta - Y^2\cos^2\theta] = 0$$ $$\tan^2\theta = {\mathbb{E}[X^2] \over \mathbb{E}[Y^2]}$$ $$\theta = \tan^{-1}\bigg[\sqrt{\mathbb{E}[X^2] \over \mathbb{E}[Y^2]}\bigg]$$