Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2001

Part 1.

1. (10 Points)

Consider the following random experiment: A fair coin is repeatedly tossed until the same outcome (H or T) appears twice in a row.

(a) What is the probability that this experiment terminates on or before the seventh coin toss?

(b) What is the probability that this experiment terminates with an even number of coin tosses?

Part 2.

2. (25 Points)

Let $\mathbf{X}$ and $\mathbf{Y}$ be independent Poisson random variables with mean $\lambda$ and $\mu$ , respectively. Let $\mathbf{Z}$ be a new random variable defined as $\mathbf{Z}=\mathbf{X}+\mathbf{Y}.$

(a) Find the probability mass function (pmf) of $\mathbf{Z}$ .

(b) Find the conditional probability mass function (pmf) of $\mathbf{X}$ conditional on the event $\left\{ \mathbf{Z}=n\right\}$ . Identify the type of pmf that this is, and fully specify its parameters.

Part 3.

3. (30 Points)

Let $\mathbf{X}_{1},\cdots,\mathbf{X}_{n},\cdots$ be a sequence of random variables that are not necessarily statistically independent, but that each have identical mean $\mu$ and variance $\sigma^{2}$ . Let $\mathbf{Y}_{1},\cdots,\mathbf{Y}_{n},\cdots$ be a sequence of random variable with $\mathbf{Y}_{n}=\frac{1}{n}\sum_{k=1}^{n}\mathbf{X}_{k}.$

(a) Given that $\mathbf{X}_{1},\cdots,\mathbf{X}_{n},\cdots$ are uncorrelated, determine whether or not $\left\{ \mathbf{Y}_{n}\right\}$ converges to $\mu$ in the mean square sense.

(b) Given that the covariance between $\mathbf{X}_{j}$ and $\mathbf{X}_{k}$ is given by
$cov\left(\mathbf{X}_{j},\mathbf{X}_{k}\right)=\begin{cases} \begin{array}{lll} \sigma^{2} \text{, for }j=k\\ r\sigma^{2} \text{, for }\left|j-k\right|=1\\ 0 \text{, elsewhere, } \end{array}\end{cases}$
where $-1\leq r\leq1$ , determine whether or not $\left\{ \mathbf{Y}_{n}\right\}$ converges to $\mu$ in the mean square sense.

Part 4.

4. (35 Points)

Let $\left\{ t_{k}\right\}$ be the set of Poisson points corresponding to a homogeneous Poisson process with parameters $\lambda$ on the real line such that if $\mathbf{N}\left(t_{1},t_{2}\right)$ is defined as the number of points in the interval $\left[t_{1},t_{2}\right)$ , then $P\left(\left\{ N\left(t_{1},t_{2}\right)=k\right\} \right)=\frac{\left[\lambda\left(t_{2}-t_{1}\right)\right]^{k}e^{-\lambda\left(t_{2}-t_{1}\right)}}{k!}\;,\qquad k=0,1,2,\cdots,\; t_{2}>t_{1}\geq0. Let \mathbf{X}\left(t\right)=\mathbf{N}\left(0,t\right)$ be the Poisson counting process for $t>0$ (note that $\mathbf{X}\left(0\right)=0$ ).

(a) Find the (first order) characteristic function of $\mathbf{X}\left(t\right)$ .

(b) Find the mean and variance of $\mathbf{X}\left(t\right)$ .

(c) Deriven an expression for the autocorrelation function of $\mathbf{X}\left(t\right)$ .

(d) Assuming that $t_{2}>t_{1}$ , find an expression for $P\left(\left\{ \mathbf{X}\left(t_{1}\right)=m\right\} \cap\left\{ \mathbf{X}\left(t_{2}\right)=n\right\} \right)$ , for all $m=0,1,2,\cdots$ and $n=0,1,2,\cdots$ .