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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 $$\begin{pmatrix} U \\ V \end{pmatrix} = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{pmatrix} \begin{pmatrix} X \\ Y \end{pmatrix}.$$

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$(X,Y)$ 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 $1$.

Note: A second approach is to use characteristic functions.

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  • $\begingroup$ Thanks, I think I just needed to look more into multivariate Gaussian variables before attempting this problem. $\endgroup$ – student0001 Apr 21 '20 at 13:06

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