Consider two independent, identically distributed random variables \(\Theta_{1}\) and \(\Theta_{2}\), each of which obeys a probability density function

\[ p_{\Theta}(\theta)=\left\{\begin{array}{cc} \frac{1}{2 \pi} & -\pi<\theta \leq \pi \\ 0 & \text { otherwise } \end{array}\right. \]

(a) Find the probability density function of the random variable \(Z\) defined by

\[ Z=\Theta_{1}+\Theta_{2} \]

(b) If \(Z\) represents a phase angle that can only be measured modulo \(2 \pi\), show that, despite the result of (a), \(Z\) is uniformly distributed on \((-\pi, \pi)\).