Line 46: Line 46:
 
==Question==
 
==Question==
 
[[Image:Q2FO12013.png|Alt text|500x500px]]
 
[[Image:Q2FO12013.png|Alt text|500x500px]]
 
[[Image:Q2FO12013D.png|Alt text|500x500px]]
 
  
 
=Solution=
 
=Solution=
[[Image:cube.png|Alt text|500x500px]]
+
[[Image:cube.png|Alt text|300x300px]]
  
 
<math>
 
<math>
Line 58: Line 56:
 
\frac{\partial}{\partial x}(2)&=0=\rho \quad \text{(no charge)}
 
\frac{\partial}{\partial x}(2)&=0=\rho \quad \text{(no charge)}
 
\end{align*}
 
\end{align*}
\vspace{1cm}
+
</math>
 +
 
 +
<math>
 
\underline{Also}:
 
\underline{Also}:
 
\begin{align*}
 
\begin{align*}

Revision as of 21:29, 24 April 2017


ECE Ph.D. Qualifying Exam

Fields and Optics (FO)

Question 1: Statics 1

August 2013



Question

Alt text

Alt text

Solution

$ \begin{equation*} \left.\begin{aligned} \nabla\cdot \bar{B}&=0\\ \oint_S \bar{B}\cdot d\bar{S}&=0 \end{aligned}\right\} \longrightarrow \Phi=\oint_S \bar{B}\cdot d\bar{S} \Longrightarrow \boxed{ \Phi=\oint_S \bar{B}\cdot d\bar{S}=0} \end{equation*} $

The magnetic flux through this closed surface is $ \Phi $

$ \begin{equation*} \boxed{\Phi=0} \end{equation*} $

Question

Alt text

Solution

Alt text

$ \begin{align*} \nabla\dot \bar{D}&=\rho \\ \quad (\frac{\partial}{\partial x}\hat{x}+\frac{\partial}{\partial y}\hat{y}+\frac{\partial}{\partial z}\hat{z})\cdot(2\hat{x})&=\rho \\ \frac{\partial}{\partial x}(2)&=0=\rho \quad \text{(no charge)} \end{align*} $

$ \underline{Also}: \begin{align*} \oint \bar{D}\cdot d\bar{S}&=Q &=\int2(dS_x)+\int2(-dS_x)=2-2=\boxed{0} \end{align*} $


Question

Part 1.

Consider $ n $ independent flips of a coin having probability $ p $ of landing on heads. Say that a changeover occurs whenever an outcome differs from the one preceding it. For instance, if $ n=5 $ and the sequence $ HHTHT $ is observed, then there are 3 changeovers. Find the expected number of changeovers for $ n $ flips. Hint: Express the number of changeovers as a sum of Bernoulli random variables.

Click here to view student answers and discussions

Part 2.

Let $ X_1,X_2,... $ be a sequence of jointly Gaussian random variables with covariance

$ Cov(X_i,X_j) = \left\{ \begin{array}{ll} {\sigma}^2, & i=j\\ \rho{\sigma}^2, & |i-j|=1\\ 0, & otherwise \end{array} \right. $

Suppose we take 2 consecutive samples from this sequence to form a vector $ X $, which is then linearly transformed to form a 2-dimensional random vector $ Y=AX $. Find a matrix $ A $ so that the components of $ Y $ are independent random variables You must justify your answer.

Click here to view student answers and discussions

Part 3.

Let $ X $ be an exponential random variable with parameter $ \lambda $, so that $ f_X(x)=\lambda{exp}(-\lambda{x})u(x) $. Find the variance of $ X $. You must show all of your work.

Click here to view student answers and discussions

Part 4.

Consider a sequence of independent random variables $ X_1,X_2,... $, where $ X_n $ has pdf

$ \begin{align}f_n(x)=&(1-\frac{1}{n})\frac{1}{\sqrt{2\pi}\sigma}exp[-\frac{1}{2\sigma^2}(x-\frac{n-1}{n}\sigma)^2]\\ &+\frac{1}{n}\sigma exp(-\sigma x)u(x)\end{align} $.

Does this sequence converge in the mean-square sense? Hint: Use the Cauchy criterion for mean-square convergence, which states that a sequence of random variables $ X_1,X_2,... $ converges in mean-square if and only if $ E[|X_n-X_{n+m}|] \to 0 $ as $ n \to \infty $, for every $ m>0 $.

Click here to view student answers and discussions

Back to ECE Qualifying Exams (QE) page

Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett