I have a wide sense stationary stochastic process x(t)=asin(2πf0t)+bcos(2πf0t) where a & b are independent gaussian random variables. How can I find the Marginal probability of x(t)?
I am confused even about the very meaning of it. How can I find marginal probability of x(t) if I don't have any other function to find its joint probability distribution with? Does it automatically imply that the other function is x(t+τ) ?
Should I just substitute x(t) as 'x' in the usual marginal gaussian probability distribution equation shown below?
The mean and variance for a,b were given as zero and 1 respectively so I calculated the autocorrelation function for x(t). Does that have any use in such scenario?
If $a$ and $b$ are independent gaussian random variables and if $a$ and $b$ are centered with the same variance $\sigma^2$ then, for every $t$, $x(t)=a\sin(2πf_0t)+b\cos(2πf_0t)$ is gaussian centered with variance $\sigma^2\sin^2(2πf_0t)+\sigma^2\cos^2(2πf_0t)=\sigma^2$. If $a$ and $b$ are centered with different variances, then $x(t)$ is gaussian centered with variance $\sigma_a^2\sin^2(2πf_0t)+\sigma_b^2\cos^2(2πf_0t)$, depending on $t$.