ECE Ph.D. Qualifying Exam

Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

January 2006


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. $


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


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


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

Click here to view student answers and discussions

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 $ .


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


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


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

Click here to view student answers and discussions

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.

Click here to view student answers and discussions

Back to ECE Qualifying Exams (QE) page

Alumni Liaison

Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.

Buyue Zhang