(Solution) - Let X and Y are statistically independent Gaussian distributed -(2025 Original AI-Free Solution)

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Let X and Y are statistically independent Gaussian-distributed random variables, each with zero mean and unit variance. Define the Gaussian process Z (t) = Z cos (2?t) + Y sin (2?t)
(a) Determine the joint probability density function of the random variables Z (t1) and Z (i2) obtained by observing Z (t) at times t= and t2 respectively.
(b) Is the process Z (t) stationary? Why?