Given two independent, wide-sense stationary random processes X(t) and Y(t) with auto correlation functions R_x(τ) and R_Y(τ), respectively.

(a) Show that the auto correlation function R_Z(τ) of their product Z(t) = X(t) Y(t) is given by

R_z(τ) = R_x(τ)R_Y(τ)

(b) Express the power spectral density of Z(t) in terms of the power spectral densities of X (t) and Y(t), denoted as S_X (f) and S_Y (f), respectively.

(c) Let X (t) be a band limited stationary noise process with power spectral density S_X (f) = 10II(f/200), and let Y(t) be the process defined by sample functions of the form

Y(t) = 5 cos(50πt + θ)

where θ is a uniformly distributed random variable in the interval (0, 2π). Using the results derived in parts (a) and (b), obtain the auto correlation function and power spectral density of Z(t) = X (t) Y(t).