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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Final Exam}
\date{May 23, 2013}
\maketitle

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% \vspace*{\fill}
% \noindent
% For questions with an easy (E) and an advanced (A) version, choose
% either the easy or the advanced version, not both.  You may make a
% different choice for each question.


% \vspace*{\fill}
% \noindent
% Useful identities:
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%   \sin \theta = \frac{\e^{\i\theta}-\e^{-\i\theta}}{2i} \\
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%   \sin^2 \theta = \tfrac12 \, (1 - \cos 2\theta) \vphantom\int \\
%   \cos \theta \, \sin \theta = \tfrac12 \, \sin 2 \theta \vphantom\int \\
%   \tilde f(\xi)
%   = \frac1{\sqrt{2\pi}} \int_{-\infty}^\infty \e^{-\i\xi x} \, f(x) \,
%   dx \qquad 
%   f(x)
%   = \frac1{\sqrt{2\pi}} \int_{-\infty}^\infty \e^{\i\xi x} \,
%   \tilde f(\xi) \, d\xi \\
%   \delta(x)
%   = \frac1{2\pi} \int_{-\infty}^\infty \e^{\i\xi x} \, d\xi  \\
%   c_k = \frac1L \int_{-L/2}^{L/2} \e^{-2 \pi \i k x/L} \, f(x) \, \d x
%   \qquad
%   f(x) = \sum_{k=-\infty}^\infty c_k \, \e^{2 \pi \i k x/L}
% \end{gather*}

% \newpage 

\begin{enumerate}

\item Let 
\[
  A =
  \begin{pmatrix}
    2 & 0 & 2 \\
    3 & 2 & 7 \\
    2 & 1 & 4
  \end{pmatrix} \,.
\]
Find a basis for $\Ker A$ and $\Range A$.
\hfill(10)


%\newpage 



\item Let $P_2$ denote the vector space of polynomials up to degree
$2$ endowed with inner product
\[
  \langle p, q \rangle
  = \frac12 \int_{-1}^1 p(x) \, q(x) \, \d x \,.
\]
and basis $B = \{ 1, x, x^2 \}$.

\begin{enumerate}
\item Find an orthogonal basis $E = \{ e_1, e_2, e_3 \}$ for $P_2$.

\emph{Note:} To simplify computations, have a careful look at which
vectors in $B$ are already orthogonal.  Normalization is not required.

\item Find the matrix $S$ representing the change of basis from
$E$ to $B$ and compute $S^{-1}$.

\item Define a linear map $L\colon V \to V$ by
\[
  (Lp)(x) = x \, p'(x) \,.
\]
Find the matrix representation of $L$ with respect to the basis $B$.

\item Find the matrix representation of $L$ with respect to the basis
$E$.

\end{enumerate}
\hfill(5+5+5+5)

%\newpage \mbox{}
%\newpage 




\item
\begin{itemize}
\item[(E)] Let $W$ be a nonempty proper subspace of $\mathbb R^n$ and
let $P$ be the orthogonal projector onto $W$.  What are the
eigenvalues of $P$?  (Explain!)  \hfill(8)
\item[(A)] Let $V$ be the vector space of continuous functions on
$\mathbb R$.  Define a linear operator $M$ by
\[
  (Mf)(x) = m(x) \, f(x)
\]
where $m \in V$ is fixed.  Under which conditions on $m$ does $M$ have
eigenvalues?  How are these eigenvalues characterized?
  \hfill(10)
\end{itemize}

%\newpage 

\item Given a $2\pi$-periodic function with complex Fourier
coefficients $c_k$, set $g(x) = f(x+a)$.  Show that $g$ has complex
Fourier coefficients $\e^{\i k a} \, c_k$.
\hfill(10)

%\newpage 

\item Compute
\[
   \int_{-\infty}^\infty x^2 \, \delta(x^3) \, \d x \,.
\]
\hfill(10)

%\newpage 

\item Consider the following communication network:
\begin{center}
\includegraphics[width=4cm]{network}
\end{center}
Assume the links between stations $A$, $B$, and $C$ may fail
independently of each other with failure probabilities
$P(AC)=\tfrac12$, $P(AB)=\tfrac13$, and $P(BC)=\tfrac14$,
respectively.

\begin{enumerate}
\item What is the probability that there is
a path in the network from station $A$ to $C$?

\item You are probing the network by sending a signal from station $A$
and find that you receive the signal at station $C$.  What is the
probability that the direct link $AC$ is working?
\end{enumerate}
\hfill(7+8)

%\newpage 


\item The \emph{exponential distribution} with parameter $\lambda>0$
is a continuous probability distribution with probability distribution
function
\[
  f(x) =
  \begin{cases}
    \lambda \, \e^{-\lambda x} & \text{for } x \geq 0 \,, \\
    0 & \text{for } x < 0 \,.
  \end{cases}
\]
\begin{enumerate}
\item Show that $f$ is indeed a probability distribution function by
checking that it is properly normalized.

\item Derive its moment generating function $M(t) = E[\e^{tX}]$.

\item Find its mean and variance.
\end{enumerate}
\hfill(5+5+5)

%\newpage 

\item There are two types of students.  Students of type $A$ report
sick only when they are actually sick.  Students of type $B$ report
sick with probability $1$ whenever there is an exam.  It is known that
the sickness rate of the general working population is $5\%$.  On
every exam, $10\%$ of students report sick.
\begin{enumerate}
\item  What is the probability that a randomly selected student
is of type $B$?
\hfill (5)
\item
\begin{itemize}
\item[(E)] What is the probability that a student who reports sick on
the day of the exam is of type $B$?
\hfill(4)
\item[(A)] A student reports sick on the midterm and on the final.
What is the probability that the student is of type $B$?  
\hfill(5)
\end{itemize}
\end{enumerate}

%\newpage 


\item Let $X_1, \dots, X_N$ be identically and independently
distributed random variables with $E[X_i] = \mu$ and $\Var [X_i] =
\sigma^2$. 
\begin{itemize}
\item[(E)] Show that
\[
  E \biggl[
      \sum_{i=1}^N \frac{X_i}N
    \biggr] = \mu
  \qquad \text{and} \qquad
  E \biggl[
      \sum_{i=1}^N \frac{(X_i-\mu)^2}N
    \biggr] = \sigma^2 \,.
\]
\hfill(8)
\item[(A)] Define the \emph{sample mean}
\[
  \mu_{\operatorname{sample}}
  = \sum_{i=1}^N \frac{X_i}N
\]
and the 
\emph{sample variance} 
\[
  \sigma^2_{\operatorname{sample}}
  = \sum_{i=1}^N \frac{(X_i-\mu_{\operatorname{sample}})^2}{N-1} \,.
\]
Show that $E[\sigma^2_{\operatorname{sample}}] = \sigma^2$.
\hfill(10)
\end{itemize}

%\newpage \mbox{}
%\newpage 

\end{enumerate}
\end{document}