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} $ .

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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\} $ .

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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.

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