# Showing that these two variables are independent and finding their distributions

Given that $$X$$ and $$Y$$ are independent $$N(0,1)$$ random variables, show that for any fixed $$\theta$$

$$U=X \cos \theta + Y \sin \theta, \ V = -X \sin \theta + Y \cos \theta$$ are independent and find their distributions.

I know that two continuous variables are independent if their joint density function is the product of their marginal density functions, but I am not sure how to find these functions (and the wording of the question suggests I should show independence before finding them). Also, I'm not sure if it's useful but I notice these equations can also be written as $$\begin{pmatrix} U \\ V \end{pmatrix} = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{pmatrix} \begin{pmatrix} X \\ Y \end{pmatrix}.$$

$$(X,Y)$$ is jointly normal. The vector equation you have written shows that $$(U,V)$$ is also jointly normal. [Any linear transformation applied to jointly normal random variables gives jointly normal random variables]. Hence you can prove that $$U$$ and $$V$$ are independent by just showing that their covariance is $$0$$. I will leave this checking to you. The distributions $$U$$ and $$V$$ are normal and they both have mean $$0$$. Variance of $$U$$ is $$\cos^{2} \theta+\sin^{2} \theta=1$$ and variance of $$V$$ is also $$1$$.