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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