finding out the probability density of a random process

Question

I have to find out the probability density function of a random process with the following specifications:z(t)= xcos(wt)-ysin(wt) where x and y are two independent gaussian random variables. Now what i am doing is expressing the above random process in the following form: z(t)=rcos(wt+A), where r= (x^2 + y^2)^1/2 and A= tan inverse of y/x. Now the random variable r has a Rayeligh probability density function and A has uniform density function.Thus i am getting expectation of z as 0. Am i in the right direction and what next should i do in order to find out the probability density function of z.

Answer 1

If $x$ and $y$ are independent Gaussian random variables, then the linear combination $$z(t)= x\cos(wt)-y\sin(wt)$$ is also Gaussian. Can you calculate the mean and variance of $z(t)$?