Mean of a random process

Question

I have this problem that I can't solve:

let $ x(t) = A cos(2 \pi ft + \theta ) $ a random process, where $ \theta $ is a variable that can take only the values $ -\pi/2 $ or $\pi/2$ with equal probability. Calculate the mean $ E[ x(t) ] $

Can anyone help me?

Answer 1

$$E(x(t)) = \sum_{\theta \in \lbrace \pm 1 \rbrace} p(\theta)x(t,\theta) = \frac{A}{2}\Big( \cos(2\pi f t + \frac{\pi}{2}) + \cos(2\pi f t - \frac{\pi}{2})\Big)$$ Using simple trignometric relations, $\cos(x + \frac{\pi}{2}) = -\sin x$ and $\cos(x - \frac{\pi}{2}) = \sin x$, we get $$E(x(t)) = \frac{A}{2}\Big( \cos(2\pi f t) - \cos(2\pi f t)\Big) = 0$$

Answer 2

@Ahmad , to determine $ E[x(t)x(t+\tau)]$ I calculated in this way, is that right?

$$ E[x(t)x(t+\tau)] = E\Big[A^2 cos(2\pi ft)cos(2\pi f(t+\tau))\Big] = \frac{A^2}{2} \Big[cos\Big(2 \pi ft - \frac{\pi}{2}\Big)cos\Big(2 \pi f(t+\tau)-\frac{\pi}{2}\Big) + cos\Big(2 \pi ft + \frac{\pi}{2}\Big)cos\Big(2 \pi f(t+\tau)+\frac{\pi}{2}\Big) \Big] $$