Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2000

## Question

Part 1.

a) The Laplacian density function is given by $f\left(x\right)=\frac{A}{2}e^{-A\left|x\right|}\text{ where }A>0.$ Determine its characteristic function.

b) Determine a bound on the probability that a RV is within two standard deviations of its mean.

Part 2.

$\mathbf{X}\left(t\right)$ is a WSS process with its psd zero outside the interval $\left[-\omega_{max},\ \omega_{max}\right]$ . If $R\left(\tau\right)$ is the autocorrelation function of $\mathbf{X}\left(t\right)$ , prove the following: $R\left(0\right)-R\left(\tau\right)\leq\frac{1}{2}\omega_{max}^{2}\tau^{2}R\left(0\right).$ (Hint: $\left|\sin\theta\right|\leq\left|\theta\right|$ ).

Part 3.

Inquiries arrive at a recorded message device according to a Poisson process of rate 15 inquiries per minute. Find the probability that in a 1-minute period, 3 inquiries arrive during the first 10 seconds and 2 inquiries arrive during the last 15 seconds.

Part 4.

A RV is given by $\mathbf{Z}=\sum_{n=0}^{8}\mathbf{X}_{n}$ where $\mathbf{X}_{n}$ 's are i.i.d. RVs with characteristic function given by $\Phi_{\mathbf{X}}\left(\omega\right)=\frac{1}{1-j\omega/2}.$

(a) Determine the characteristic function of $\mathbf{Z}$ .

(b) Determine the pdf of $\mathbf{Z}$ . You can leave your answer in integral form.