Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2003
5. (25% of Total)
Suppose that “sensor nodes” are spread around the ground (two-dimensional space) according to a Poisson Process, with an average density of nodes per unit area of $ \lambda $ . We are interested in the number and location of nodes inside a circle $ C $ of radius one that is centered at the origin. You must quote, but do not have to prove, properties of the Poisson process that you use in your solutions to the following questions:
a. (10%)
Given that a node is in the circle C , determine the density or distribution function of its distance $ \mathbf{D} $ from the origin.
• $ i)\; d<0,\; F_{\mathbf{D}}\left(d\right)=0. $
• $ ii)\;0\leq d\leq1,\; F_{\mathbf{D}}\left(d\right)=P\left(\left\{ \mathbf{D}\leq d\right\} \right)=\frac{d^{2}\pi}{1^{2}\pi}=d^{2}. $
• $ iii)\;1<d,\; F_{\mathbf{D}}\left(d\right)=1. $
$ \therefore\; F_{\mathbf{D}}\left(d\right)=\begin{cases} \begin{array}{lll} 0 & & ,\; d<0\\ d^{2} & & ,\;0\leq d\leq1\\ 1 & & ,\;1<d \end{array}\end{cases} $
$ f_{\mathbf{D}}\left(d\right)=2d\cdot\mathbf{1}_{\left[0,1\right]}\left(d\right). $
b. (15%)
Find the density or distribution of the distance from the center of $ C $ to the node inside $ C $ that is closest to the origin.
$ \mathbf{X} $ : the random variable for the distance from the center of $ C $ to the node inside $ C $ that is closest to the origin
$ F_{\mathbf{X}}\left(x\right)=P\left(\left\{ \mathbf{X}\leq x\right\} \right)=1-P\left(\left\{ \mathbf{X}>x\right\} \right) $$ =1-P\left(\left\{ \text{no sensor nodes in the circle that has a radius }x\right\} \right) $
$ =1-e^{-\lambda x^{2}\pi}. $
$ f_{\mathbf{X}}\left(x\right)=\frac{d}{dx}F_{\mathbf{X}}\left(x\right)=2\lambda x\pi\cdot e^{-\lambda x^{2}\pi}. $
