ECE Ph.D. Qualifying Exam

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.


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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) $ .


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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] $ .


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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.


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