ECE Ph.D. Qualifying Exam

Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

January 2003



Question

Problem 1 (30 points)

i)

Let $ \mathbf{X} $ and $ \mathbf{Y} $ be jointly Gaussian (normal) distributed random variables with mean $ 0 $ , $ E\left[\mathbf{X}^{2}\right]=E\left[\mathbf{Y}^{2}\right]=\sigma^{2} $ and $ E\left[\mathbf{XY}\right]=\rho\sigma^{2} $ with $ \left|\rho\right|<1 $ . Find the joint characteristic function $ E\left[e^{i\left(h_{1}\mathbf{X}+h_{2}\mathbf{Y}\right)}\right] $ .

ii)

Let $ \mathbf{X} $ and $ \mathbf{Y} $ be two jointly Gaussian distributed r.v's with identical means and variances but are not necessarily independent. Show that the r.v. $ \mathbf{V}=\mathbf{X}+\mathbf{Y} $ is independeent of the r.v. $ \mathbf{W}=\mathbf{X}-\mathbf{Y} $ . Is the same answer true for $ \mathbf{A}=f\left(\mathbf{V}\right) $ and $ \mathbf{B}=g\left(\mathbf{W}\right) $ where $ f\left(\cdot\right) $ and $ g\left(\cdot\right) $ are suitable functions such that $ E\left[f\left(\mathbf{V}\right)\right]<\infty $ and $ E\left[g\left(\mathbf{W}\right)\right]<\infty $ . Given reasons.

iii)

Let $ \mathbf{X} $ and $ \mathbf{Y} $ be independent $ N\left(m,1\right) $ random variables. Show that the sample mean $ \mathbf{M}=\frac{\mathbf{X}+\mathbf{Y}}{2} $ is independent of the sample variance $ \mathbf{V}=\left(\mathbf{X}-\mathbf{M}\right)^{2}+\left(\mathbf{Y}-\mathbf{M}\right)^{2} $ .

Click here to view student answers and discussions

Problem 2 (35 points)

Consider the stochastic process $ \left\{ \mathbf{X}_{n}\right\} $ defined by: $ \mathbf{X}_{n+1}=a\mathbf{X}_{n}+b\mathbf{W}_{n} where \mathbf{X}_{0}\sim N\left(0,\sigma^{2}\right) $ and $ \left\{ \mathbf{W}_{n}\right\} $ is an i.i.d. $ N\left(0,1\right) $ sequence of r.v's independent of $ \mathbf{X}_{0} $ .

i)

Show that if $ R_{k}=cov\left(\mathbf{X}_{k},\mathbf{X}_{k}\right) $ converges as $ k\rightarrow\infty $ , then $ \left\{ \mathbf{X}_{k}\right\} $ converges to a w.s.s. process.

ii)

Show that if $ \sigma^{2} $ is chosen appropriately and $ \left|a\right|<1 $ , then $ \left\{ \mathbf{X}_{k}\right\} $ will be a stationary process for all $ k $ .

iii)

If $ \left|a\right|>1 $ , show that the variance of the process $ \left\{ \mathbf{X}_{k}\right\} $ diverges but $ \frac{\mathbf{X}_{k}}{\left|a\right|^{k}} $ converges in the mean square.

Click here to view student answers and discussions

Problem 3 (35 points)

i)

Catastrophes occur at times $ \mathbf{T}_{1},\mathbf{T}_{2},\cdots $, where $ \mathbf{T}_{i}=\sum_{k=1}^{i}\mathbf{X}_{k} $ where the $ \mathbf{X}_{k} $ 's are independent, identically distributed positive random variables. Let $ \mathbf{N}_{t}=\max\left\{ n:\mathbf{T}_{n}\leq t\right\} $ be the number of catastrophes which have occurred by time $ t $ . Show that if $ E\left[\mathbf{X}_{1}\right]<\infty $ then $ \mathbf{N}_{t}\rightarrow\infty $ almost surely (a.s.) and $ \frac{\mathbf{N}_{t}}{t}\rightarrow\frac{1}{E\left[\mathbf{X}_{1}\right]} $ as $ t\rightarrow\infty $ a.s.

ii)

Let $ \left\{ \mathbf{X}_{t},t\geq0\right\} $ be a stochastic process defined by: $ \mathbf{X}_{t}=\sqrt{2}\cos\left(2\pi\xi t\right) $ where $ \xi $ is a $ N\left(0,1\right) $ random variable. Show that as $ t\rightarrow\infty,\;\left\{ \mathbf{X}_{t}\right\} $ converges to a wide sense stationary process. Find the spectral density of the limit process.

Hint:

Use the fact that the characteristic function of a $ N\left(0,1\right) $ r.v. is given by $ E\left[e^{ih\mathbf{X}}\right]=e^{-\frac{h^{2}}{2}} $ .

Click here to view student answers and discussions


Back to ECE Qualifying Exams (QE) page

Alumni Liaison

Prof. Math. Ohio State and Associate Dean
Outstanding Alumnus Purdue Math 2008

Jeff McNeal