Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2004

## Question

1. (20 pts.)

A probability space $\left(\mathcal{S},\mathcal{F},\mathcal{P}\right)$ has a sample space consisting of all pairs of positive integers: $\mathcal{S}=\left\{ \left(k,m\right):\; k=1,2,\cdots;\; m=1,2,\cdots\right\}$ . The event space $\mathcal{F}$ is the power set of $\mathcal{S}$ , and the probability measure $\mathcal{P}$ is specified by the pmf $p\left(k,m\right)=p^{2}\left(1-p\right)^{k+m-2},\qquad p\in\left(0,1\right)$.

(a)

Find $P\left(\left\{ \left(k,m\right):\; k\geq m\right\} \right)$ .

(b)

Find $P\left(\left\{ \left(k,m\right):\; k+m=r\right\} \right)$ , for $r=2,3,\cdots$ .

(c)

Find $P\left(\left\{ \left(k,m\right):\; k\text{ is an odd number}\right\} \right)$ .

2. (20 pts.)

Let $\mathbf{X}$ and $\mathbf{Y}$ be two independent identically distributed exponential random variables having mean $\mu$ . Let $\mathbf{Z}=\mathbf{X}+\mathbf{Y}$ . Find $f_{\mathbf{X}}\left(x|\mathbf{Z}=z\right)$ , the conditional pdf of $\mathbf{X}$ given the event $\left\{ \mathbf{Z}=z\right\}$ .

3. (25 pts.)

Let $\mathbf{X}_{1},\cdots,\mathbf{X}_{n}$ be independent identically distributed (i.i.d. ) random variables uniformaly distributed over the interval $\left[0,1\right]$ .

(a)

Find the probability density function of $\mathbf{Y}=\max\left\{ \mathbf{X}_{1},\cdots,\mathbf{X}_{n}\right\}$ .

(b)

Find the probability density function of $\mathbf{Z}=\min\left\{ \mathbf{X}_{1},\cdots,\mathbf{X}_{n}\right\}$ .

4. (35 pts.)

Assume that $\mathbf{X}\left(t\right)$ is a zero-mean, continuous-time, Gaussian white noise process with autocorrelation function $R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=\frac{N_{0}}{2}\delta\left(t_{1}-t_{2}\right).$ Let $\mathbf{Y}\left(t\right)$ be a new random process defined as the output of a linear time-invariant system with impulse response $h\left(t\right)=\frac{1}{T}e^{-t/T}\cdot u\left(t\right),$ where $u\left(t\right)$ is the unit step function and $T>0$ .

(a)

What is the mean of $\mathbf{Y\left(t\right)}$ ?

(b)

What is the autocorrelation function of $\mathbf{Y}\left(t\right)$ ?

(c)

Write an expression for the $n$ -th order characteristic function of $\mathbf{Y}\left(t\right)$ sampled at time $t_{1},t_{2},\cdots,t_{n}$ . Simplify as much as possible.

(d)

Write an expression for the second-order pdf $f_{\mathbf{Y}\left(t_{1}\right)\mathbf{Y}\left(t_{2}\right)}\left(y_{1},y_{2}\right)$ of $\mathbf{Y}\left(t\right)$ . simplify as much as possible.

(e)

Find the minium mean-square estimate of $\mathbf{Y}\left(t_{2}\right)$ given that $\mathbf{Y}\left(t_{1}\right)=y_{1}$ . Simplify your answer as much as possible.