Let the real-valued random process \(U(t)\) be defined by

\[ U(t)=A \cos (2 \pi v t-\Phi) \]

where \(v\) is a known constant, \(\Phi\) is a random variable uniformly distributed on \((-\pi, \pi), A\) is a random variable that takes on the values 1 and 2 with equal probabilities of \(1 / 2\), and \(A\) and \(\Phi\) are statistically independent.

(a) Calculate the time average \(\left\langle u^{2}(t)\rightangle\) for a sample function with amplitude 1 and a sample function with amplitude 2.

(b) Calculate \(\overline{u^{2}}\).

(c) Show that \[ \overline{u^{2}}=\frac{1}{2}\left\langle u^{2}(t)\rightangle_{1}+\frac{1}{2}\left\langle u^{2}(t)\rightangle_{2} \]
where \(\left\langle u^{2}(t)\rightangle_{1}\) and \(\left\langle u^{2}(t)\rightangle_{2}\) are the results of part (a) for amplitudes of 1 and 2 , respectively.