Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2005

3. (40 Points)

Consider a homogeneous Poisson point process with rate $\lambda$ and points (event occurrence times) $\mathbf{T}_{1},\mathbf{T}_{2},\cdots,\mathbf{T}_{n},\cdots$ .

(a)

Derive the pdf $f_{k}\left(t\right)$ of the $k$ -th point $\mathbf{T}_{k}$ for arbitrary $k$ .

• The cdf is $F_{\mathbf{T}_{k}}\left(t\right)=P\left(\mathbf{T}_{k}\leq t\right)=P\left(\text{at least }k\text{ points within }t\right)=\sum_{j=k}^{\infty}\frac{e^{-\lambda t}\cdot\left(\lambda t\right)^{j}}{j!} $$=1-P\left(\mathbf{T}_{k}>t\right)=1-P\left(\text{less than }k\text{ points within }t\right)=1-\sum_{j=0}^{k-1}\frac{e^{-\lambda t}\cdot\left(\lambda t\right)^{j}}{j!}. • The pdf by differentiating the cdf isf_{\mathbf{T}_{k}}\left(t\right)=\frac{dF_{\mathbf{T}_{k}}}{dt}=-\sum_{j=0}^{k-1}\frac{\left(-\lambda\right)e^{-\lambda t}\cdot\left(\lambda t\right)^{j}}{j!}-\sum_{j=0}^{k-1}\frac{e^{-\lambda t}\cdot j\left(\lambda t\right)^{j-1}\cdot\lambda}{j!}$$ =\lambda e^{-\lambda t}\sum_{j=0}^{k-1}\frac{\left(\lambda t\right)^{j}}{j!}-\lambda e^{-\lambda t}\sum_{j=1}^{k-1}\frac{j\left(\lambda t\right)^{j-1}}{j!}=\lambda e^{-\lambda t}\sum_{j=0}^{k-1}\frac{\left(\lambda t\right)^{j}}{j!}-\lambda e^{-\lambda t}\sum_{j=1}^{k-1}\frac{\left(\lambda t\right)^{j-1}}{\left(j-1\right)!} $$=\lambda e^{-\lambda t}\sum_{j=0}^{k-1}\frac{\left(\lambda t\right)^{j}}{j!}-\lambda e^{-\lambda t}\sum_{j=0}^{k-2}\frac{\left(\lambda t\right)^{j}}{j!}=\lambda e^{-\lambda t}\frac{\left(\lambda t\right)^{k-1}}{\left(k-1\right)!}. – This is Erlang distribution. (b) What kind of distribution does \mathbf{T}_{1} have? • If k=1 , then f_{\mathbf{T}_{1}}\left(t\right)=\lambda e^{-\lambda t} . Thus \mathbf{T}_{1} is a exponential random variable with parameter \lambda . (c) What is the conditional pdf of \mathbf{T}_{k} given \mathbf{T}_{k-1}=t_{0} , where t_{0}>0 ? (You can give the answer without derivation if you know it.) • The conditional cdf is F_{\mathbf{T}_{k}}\left(t_{k}|\mathbf{T}_{k-1}=t_{0}\right)=P\left(\mathbf{T}_{k}\leq t_{k}|\mathbf{T}_{k-1}=t_{0}\right)=P\left(N\left(t_{k},t_{0}\right)\geq1\right)$$ =1-P\left(N\left(t_{k},t_{0}\right)=0\right)=1-e^{-\lambda\left(t_{k}-t_{0}\right)}.$

• The conditional pdf by differentiating the conditional cdf is

$\therefore f_{\mathbf{T}_{k}}\left(t_{k}|\mathbf{T}_{k-1}=t_{0}\right)=\lambda e^{-\lambda\left(t_{k}-t_{0}\right)}.$

(d)

Suppose you have a random number generator that produces independent, identically distributed (i.i.d. ) random variables $\mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n},\cdots$ that are uniformaly distributed on the interval $\left(0,1\right)$ . Explain how you could use these to simulate the Poisson points $\mathbf{T}_{1},\mathbf{T}_{2},\cdots,\mathbf{T}_{n},\cdots$ describe above. Provide as complete an explanation as possible.

• This problem is similar to QE 2003 August Problem 2.