A certain random process \(U(t)\) takes on equally probable values +1 or 0 with changes occurring randomly in time. The probability that \(n\) changes occur in time \(\tau\) is known to be

\[ P_{N}(n)=\frac{1}{1+\alpha|\tau|}\left(\frac{\alpha|\tau|}{1+\alpha|\tau|}\right)^{n}, \quad n=0,1,2, \ldots \]

(a) Show that the mean value of \(n\) is \(\alpha|\tau|\).

(b) Find and sketch the autocorrelation function of \(U(t)\).

\(\quad \sum_{k=0}^{\infty} r^{k}=\frac{1}{1-r}\) when \(|r|<1\).