Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

January 2006

## Question

1 (33 points)

Let $\mathbf{X}$ and $\mathbf{Y}$ be two joinly distributed random variables having joint pdf

$f_{\mathbf{XY}}\left(x,y\right)=\left\{ \begin{array}{lll} 1, & & \text{ for }0\leq x\leq1\text{ and }0\leq y\leq1\\ 0, & & \text{ elsewhere. } \end{array}\right.$

(a)

Are $\mathbf{X}$ and $\mathbf{Y}$ statistically independent? Justify your answer.

(b)

Let $\mathbf{Z}$ be a new random variable defined as $\mathbf{Z}=\mathbf{X}+\mathbf{Y}$ . Find the cdf of $\mathbf{Z}$ .

(c)

Find the variance of $\mathbf{Z}$ .

2 (33 points)

Suppose that $\mathbf{X}$ and $\mathbf{N}$ are two jointly distributed random variables, with $\mathbf{X}$ being a continuous random variable that is uniformly distributed on the interval $\left(0,1\right)$ and $\mathbf{N}$ being a discrete random variable taking on values $0,1,2,\cdots$ and having conditional probability mass function $p_{\mathbf{N}}\left(n|\left\{ \mathbf{X}=x\right\} \right)=x^{n}\left(1-x\right),\quad n=0,1,2,\cdots$ .

(a)

Find the probability that \mathbf{N}=n .

(b)

Find the conditional density of $\mathbf{X}$ given $\left\{ \mathbf{N}=n\right\}$ .

(c)

Find the minimum mean-square error estimator of $\mathbf{X}$ given $\left\{ \mathbf{N}=n\right\}$ .

3 (34 points)

Assume that the locations of cellular telephone towers can be accurately modeled by a 2-dimensional homogeneous Poisson process for which the following two facts are know to be true:

1. The number of towers in a region of area A is a Poisson random variable with mean \lambda A , where \lambda>0 .

2. The number of towers in any two disjoint regions are statistically independent.

Assume you are located at a point we will call the origin within this 2-dimensional region, and let $R_{\left(1\right)}<R_{\left(2\right)}<R_{\left(3\right)}<\cdots$ be the ordered distances between the origin and the towers.

(a) Show that $R_{\left(1\right)}^{2},R_{\left(2\right)}^{2},R_{\left(3\right)}^{2},\cdots$ are the points of a one-dimensional homogeneous Poisson process.

(b) What is the rate of the Poisson process in part (a)? $\lambda\pi$ .

(c) Determine the density function of $R_{\left(k\right)}$ , the distance to the $k$ -th nearest cell tower.