I was solving the following question and I derived the Auto correlation function and proved that it is a WSS process. However, I am not sure how to go about finding the Marginal probability density function. Should I find joint probability of a and b and then find marginal probability of each random variable?
Consider the stochastic process: $x(t) = a \sin(2\pi f_0 t) + b \cos(2\pi f_0 t)$ where $f_0$ is given, a and b are assumed to be uncorrelated Gaussian random variables, each having zero mean and unit standard deviation. Is $x(t)$ a wide sense stationary stochastic process? Find the marginal density function and auto-correlation function of the random process $x(t)$.
At each moment of time, $x_t$ is a linear combination of two independent Gaussian random variables $a$ and $b$, so it is also a Guassian random variable. Recall that if $a,b\sim \mathcal N(0,1)$ then $\alpha a + \beta b\in \mathcal N(0,a^2+b^2)$.