Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2005

## Question

1. (30 Points)

Assume that $\mathbf{X}$ is a binomial distributed random variable with probability mass function (pmf) given by $p_{n}\left(k\right)=\left(\begin{array}{c} n\\ k \end{array}\right)p^{k}\left(1-p\right)^{n-k}\;,\qquad k=0,1,2,\cdots,n$ where $0<p<1$ .

(a)

Find the characteristic function of $\mathbf{X}$ . (You must show how you derive the characteristic function.)

(b)

Compute the standard deviation of $\mathbf{X}$ .

(c)

Find the value or values of $k$ for which $p_{n}\left(k\right)$ is maximum, and express the answer in terms of $p$ and $n$ . Give the most complete answer to this question that you can.

2. (30 Points)

Let $\mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n},\cdots$ be a sequence of binomially distributed random variables, with $\mathbf{X}_{n}$ having probability mass function $p_{n}\left(k\right)=\left(\begin{array}{c} n\\ k \end{array}\right)p_{n}^{k}\left(1-p_{n}\right)^{n-k}\;,\qquad k=0,1,2,\cdots,n,$ where $0<p_{n}<1$ for all $n=1,2,3,\cdots$ . Show that if $np_{n}\rightarrow\lambda\text{ as }n\rightarrow\infty,$ then the random sequence $\mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n},\cdots$ converges in distribution to a Poisson random variable having mean $\lambda$ .

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 $k$ .

(b)

What kind of distribution does $\mathbf{T}_{1}$ have?

(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.)

(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.