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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Final Exam}
\date{May 24, 2003}
\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
%% Fourier Series for a periodic function on the interval $[a,a+L]$:
%% \begin{gather*}
%%   f_k
%%   = \frac1{\sqrt{L}} \int_a^{a+L} e^{-2 \pi i k x/L} \, f(x) \,
%%   dx \,, \qquad
%%   f(x) = \frac1{\sqrt{L}} \sum_{k=-\infty}^\infty f_k \, e^{2 \pi i
%%   k x/L} 
%% \end{gather*}
%% \noindent Fourier Transform:
%% \begin{gather*}
%%   \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 \\
%% \end{gather*}
%% Also note:
%% \begin{gather*}
%%   \sum_{n=0}^\infty x^n = \frac1{1-x} \qquad \text{when } |x|<1
%% \end{gather*}

%% \newpage


\begin{enumerate}

\item 
\begin{enumerate}
\item Compute the inverse of the Matrix
\[
  A = \begin{pmatrix}
        i & 0 & -i \\
        i & -i & 0 \\
        0 & i & i
      \end{pmatrix} \,.
\]

\item Using the result from part (a), or otherwise, solve the system
of linear equations
\[
  A \, \x = \begin{pmatrix} i \\ 1 \\ -i \end{pmatrix} \,.
\]
\end{enumerate}
\hfill (10+5)
%% \newpage

\item
\begin{itemize}
\item[(E)] Find the matrix representing---with respect to the standard
basis---the projection in $\R^2$ onto the vector
\[
  \v = 
  \begin{pmatrix}
  1 \\ -1 
  \end{pmatrix} \,.
\]
\hfill (4)

\item[(A)] Let $V$ be a real vector space with an inner product,
$\{\e_1, \dots, \e_n \}$ an orthonormal basis for $V$, and $T$ a
linear transformation on $V$.  Explain why the matrix $S$ which
represents $T$ with respect to the basis $\{\e_1, \dots, \e_n \}$ has
the components
\[
  s_{ij} = \langle T(\e_j), \e_i \rangle \,.
\]
\hfill (5)
%% \newpage

\end{itemize}

\item Consider the vector space $P_2$ of polynomials of degree less or
equal to $2$ with basis
\[
  B = \{ 1, x, x^2 \} \,.
\]

\begin{enumerate}
\item What is the matrix $S$ representing the derivative on $P_2$ with
respect to the basis $B$?

\item Find the eigenvalues and eigenvectors of $S$.

\item Show that $S$ is not diagonalizable.  Is there another basis for
$P_2$ such that the derivative with respect to this other basis is
diagonalizable?  Explain.

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

\item Compute the complex Fourier series for the function $f(x)=x$ on
the interval $[-\pi,\pi]$.
\hfill (10)
%% \newpage

\item
\begin{itemize}
\item[(E)] Show that
\[
  \mathcal{F} \left( \frac{df}{dx} \right) = i\xi \, \mathcal{F}(f)
  \,,
\]
where $\mathcal{F}$ denotes the Fourier transform.
\hfill(8)

\item[(A)] It is known that the Fourier transform of a Gaussian
distribution is again a (non-normalized) Gaussian distribution.  More
precisely, if $\phi_\sigma$ denotes the Gaussian with mean zero and
variance $\sigma^2$, i.e.
\[
  \phi_\sigma(x) = \frac1{\sigma\sqrt{2\pi}} \,
  e^{-\tfrac{x^2}{2 \sigma^2}} \,,
\]
then
\[
  \tilde \phi_\sigma(\xi) = \frac1{\sqrt{2 \pi}} \,
  e^{-\tfrac{\sigma^2 \xi^2}{2}} \,.
\]
Show that the convolution of two Gaussian distributions $\phi_\sigma$
and $\phi_\mu$ is again a Gaussian.  What is the variance of the
resulting Gaussian?
\hfill (10)
\end{itemize}
%% \newpage

\item 

\begin{itemize}
\item[(E)] Compute
\[
  \int_{-\infty}^\infty \delta (x^{1/3}) \, dx \,.
\]
\hfill(8)

\item[(A)] If $X$ is a continuous random variable with probability
density function $p(x)$, and $Y=g(X)$ another random variable, then
the probability density function of $Y$ is given by
\[
  q(y) = \int_{-\infty}^\infty p(x) \, \delta (g(x) - y) \, dx \,.
\]
Compute $q(y)$ when $g(x)=x^{1/3}$, and $p(x) = 1$ on the interval
$[0,1]$ and $p(x)=0$ otherwise.
\end{itemize}
\hfill(10)
%% \newpage

\item Each question on a multiple choice exam has three possible
answers, of which only one is correct.  Some student did not study
very hard and there is only a 50\% chance that he knows the anwer to a
particular question.  Of course he will select an answer at random in
case he does not know the correct one.  What is the probability that
he knows the answer to a question he has answered correctly?
\hfill(10)

%% \newpage

\item A homework set in ESM 2B has six questions.  For the
corresponding quiz, three of these questions are selected at random,
each selection being independent of the others.

\begin{itemize}
\item[(E)] What is the probability that you and your best friend get
an identical quiz?

\hfill(8)

\item[(A)] Find a formula for the probability that in a group of $k$
students at least two get an identical quiz.
\hfill(10)

\end{itemize}
%% \newpage

\item An experiment is independently repeated $n$ times.  The
experiment has two possible outcomes: ``success'' with probability $p$
and ``failure'' with probability $q=1-p$.  Consider the random
variable
\[
  X = \text{number of trials required to obtain the first success}\,.
\]
\begin{enumerate}
\item Explain why the probability function of $X$ is
\[
  P(X=k) = p \, q^{k-1} \,.
\]

\item Show that the moment generating function $M_X(t)=E[e^{tX}]$ is
given by
\[
  M_X(t) = \frac{p \, e^t}{1 - q \, e^{t}} \,.
\]
\item Compute, by using the moment generating function, or otherwise,
the mean and the variance of $X$.

\end{enumerate}
\hfill(5+5+5)
%% \newpage\null
%% \newpage\null
\end{enumerate}
\end{document}
