Problem 5.28 A mathematical model for a random sine wave X(t) assumes that X(t) = A cos(t + ) where A (the amplitude) and

Problem 5.28 A mathematical model for a random sine wave X(t) assumes that X(t) = A cos(t + Θ) where A (the amplitude) and Θ (the phase) are mutually independent random variables, Θ is uniformly distributed over [0, 2π], and E(A) = µ1 and E(A2) = µ2 are both finite.

(a) Show that E(X(t)) = 0 for every t.

(b) Show that Cov(X(t1), X(t2)) = (µ2/2) cos(t1 − t2). Hint: cos(φ1) cos(φ2) = (1/2)(cos(φ1 + φ2) + cos(φ1 − φ2)).