Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2006

## Question

1

Let $\mathbf{U}_{n}$ be a sequence of independent, identically distributed zero-mean, unit-variance Gaussian random variables. The sequence $\mathbf{X}_{n}$ , $n\geq1$ , is given by $\mathbf{X}_{n}=\frac{1}{2}\mathbf{U}_{n}+\left(\frac{1}{2}\right)^{2}\mathbf{U}_{n-1}+\cdots+\left(\frac{1}{2}\right)^{n}\mathbf{U}_{1}.$

(a) (15 points)

Find the mean and variance of $\mathbf{X}_{n}$ .

(b) (15 points)

Find the characteristic function of $\mathbf{X}_{n}$ .

(c) (10 points)

Does the sequence $\mathbf{X}_{n}$ converge in distribution? A simple yes or no answer is not sufficient. You must justify your answer.

2

Let $\Phi$ be the standard normal distribution, i.e., the distribution function of a zero-mean, unit-variance Gaussian random variable. Let $\mathbf{X}$ be a normal random variable with mean $\mu$ and variance 1 . We want to find $E\left[\Phi\left(\mathbf{X}\right)\right]$ .

(a) (10 points)

First show that $E\left[\Phi\left(\mathbf{X}\right)\right]=P\left(\mathbf{Z}\leq\mathbf{X}\right)$ , where $\mathbf{Z}$ is a standard normal random variable independent of $\mathbf{X}$ . Hint: Use an intermediate random variable $\mathbf{I}$ defined as

(b) (10 points)

Now use the result from Part (a) to show that $E\left[\Phi\left(\mathbf{X}\right)\right]=\Phi\left(\frac{\mu}{\sqrt{2}}\right)$ .

3 (15 points)

Let $\mathbf{Y}(t)$ be the output of linear system with impulse response $h\left(t\right)$ and input $\mathbf{X}\left(t\right)+\mathbf{N}\left(t\right)$ , where $\mathbf{X}\left(t\right)$ and $\mathbf{N}\left(t\right)$ are jointly wide-sense stationary independent random processes. If $\mathbf{Z}\left(t\right)=\mathbf{X}\left(t\right)-\mathbf{Y}\left(t\right)$ , find the power spectral density $S_{\mathbf{Z}}\left(\omega\right)$ in terms of $S_{\mathbf{X}}\left(\omega\right) , S_{\mathbf{N}}\left(\omega\right) , m_{\mathbf{X}}=E\left[\mathbf{X}\right]$ , and $m_{\mathbf{Y}}=E\left[\mathbf{Y}\right]$ .

4

Suppose customer orders arrive according to an i.i.d. Bernoulli random process $\mathbf{X}_{n}$ with parameter $p$ . Thus, an order arrives at time index $n$ (i.e., $\mathbf{X}_{n}=1$ ) with probability $p$ ; if an order does not arrive at time index $n$ , then $\mathbf{X}_{n}=0$ . When an order arrives, its size is an exponential random variable with parameter $\lambda$ . Let $\mathbf{S}_{n}$ be the total size of all orders up to time $n$ .

(a) (20 points)

Find the mean and autocorrelation function of $\mathbf{S}_{n}$ .

(b) (5 points)

Is $\mathbf{S}_{n}$ a stationary random process? Explain.