Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2002
4. (25 Points)
Assume that the distribution of stars within a galaxy is accurately modeled by a 3-dimensional homogeneous Poisson process for which the following two facts are known to be true:
• The number of starts in a region of volume $ V $ is a Poisson random variable with mean $ \lambda V $ , where $ \lambda>0 $ .
• The number of starts in any two disjoint regions are statistically independent.
Assume you are located at an arbitrary position near the center of the galaxy.
(a)
Find the probability density function (pdf) of the distance to the nearest star.
Let $ \mathbf{R} $ be the distance to nearest star.
$ F_{\mathbf{R}}\left(r\right)=P\left(\left\{ \mathbf{R}\leq r\right\} \right). $
• $ i)\; r<0,\; F_{\mathbf{R}}\left(r\right)=0. $
• $ ii)\; r\geq0, F_{\mathbf{R}}\left(r\right)=P\left(\left\{ \text{there exist one or more stars in the sphere of radius }r\right\} \right) $$ =1-P\left(\left\{ \text{no star exists in the sphere of radius }r\right\} \right) $$ =1-e^{-\frac{4}{3}\pi r^{3}\lambda}. $
$ \therefore f_{\mathbf{R}}\left(r\right)=\begin{cases} \begin{array}{lll} 4\pi r^{2}\lambda e^{-\frac{4}{3}\pi r^{3}\lambda} & & ,\; r\geq0\\ 0 & & ,\; r<0. \end{array}\end{cases} $
(b)
Find the most likely distance to the nearest star.
$ \frac{df_{\mathbf{R}}\left(r\right)}{dr}=8\pi r\lambda e^{-\frac{4}{3}\pi r^{3}\lambda}-\left(4\pi r^{2}\lambda\right)^{2}e^{-\frac{4}{3}\pi r^{3}\lambda} = 0 $
$ e^{-\frac{4}{3}\pi r^{3}\lambda}\left(8\pi r\lambda-\left(4\pi r^{2}\lambda\right)^{2}\right) = 0 $
$ 8\pi r\lambda-16\pi^{2}r^{4}\lambda^{2} = 0 $
$ 1-8\pi r^{3}\lambda = 0 $
$ \therefore r=\left(\frac{1}{2\pi\lambda}\right)^{\frac{1}{3}}. $
