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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 $x(t)$.

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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 $\alpha a + \beta b\in \mathcal N(0,a^2+b^2)$.

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  • $\begingroup$ I was able to show that it is a wide sense stationary process. I couldn't really get your answer on marginal probability density function. Could you kindly elaborate a little. $\endgroup$ – CRG Nov 20 '14 at 8:59
  • $\begingroup$ @crg123 Can you put the solution of stationarity in your post? $\endgroup$ – Ilya Nov 20 '14 at 9:01
  • $\begingroup$ Ok, did that. Now can you elaborate. $\endgroup$ – CRG Nov 20 '14 at 9:24
  • $\begingroup$ @crg123: done${}$ $\endgroup$ – Ilya Nov 20 '14 at 9:39

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