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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Final Exam}
\date{May 21, 2004}
\maketitle

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\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.

\medskip
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\noindent
Useful identities:
\begin{gather*}
  \cos \theta = \frac{e^{i\theta}+e^{-i\theta}}{2}\,, \qquad
  \sin \theta = \frac{e^{i\theta}-e^{-i\theta}}{2i} \\
  \cos^2 \theta = \tfrac12 \, (1 + \cos 2\theta) \,, \qquad
  \sin^2 \theta = \tfrac12 \, (1 - \cos 2\theta) \\
 f_k = \frac1{\sqrt{L}} \int_{a}^{a+L} \e^{-\frac{2\pi i k x}L} \, f(x)
  \, dx \,, \qquad f(x) = \frac1{\sqrt{L}} \sum_{k=-\infty}^\infty f_k \,
         \e^{\frac{2\pi i k x}L} \\
  \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\\
  f(x) = \frac1{\sqrt{2\pi} \sigma} \, \mathrm{e}^{-\tfrac{(x-\mu)^2}{2
      \sigma^2}}  \\
  P(A \cap B)=P(A) P(B|A) \\
  P(A)= P(B) P(A|B)+ P(\overline{B}) P(A|\overline{B})
\end{gather*}


%\newpage

\begin{enumerate}
\item Compute the inverse of 
  \begin{align*}
    A = \begin{pmatrix}
        \rule[-3mm]{0cm}{4mm} \frac 1 3 & \frac 2 3 & \frac 2 3 \\
        \rule[-3mm]{0cm}{4mm} \frac 2 3 & -\frac 2 3 & \frac 1 3 \\
        \rule[-3mm]{0cm}{4mm} \frac 2 3 & \frac 1 3 & -\frac 2 3
      \end{pmatrix} \,.
  \end{align*}
\hfill(10)

%\newpage
  
\item 
  Let $V$ be the vector space of continuous functions on $\R$
  spanned by the basis
  \[
  E = \{ e^{ikx}, e^{-ikx} \}
  \]
  with the usual addition and scalar multiplication.
  \begin{enumerate}
  \item Let
    \[
    T = \{ \sin kx, \cos kx \} \,.
    \]
    Explain why $T$ is also a basis of $V$, and find the matrix $S$ for
    the change of basis from $T$ to $E$.
    
    (Recall that $e^{ix}=\cos x + i \sin x$.)
    
  \item Let $f$ be defined by the Fourier sine and cosine series
    \[
      f(x) = \sum_{k=1}^\infty
        \biggl(
          2 \, \frac{\cos kx}{k} + 2 \, \frac{sin{kx}}{k}
        \biggr)\,.
    \]
    Use part (a) to compute the complex Fourier series for $f$.
  \end{enumerate}

      
  \hfill(5+5)

%\newpage

\item 
  \begin{itemize}
  \item[(E)] Show that the eigenvectors corresponding to distinct
    eigenvalues of a Hermitian matrix are orthogonal.
    \hfill(8)
  \item[(A)] Let $A$ be a Hermitian matrix and $\u $ a vector with
    $||\u||=1$. Show that
    \begin{align*}
      \lambda_{\mbox{\scriptsize min}} \le \u^H A \u \le
      \lambda_{\mbox{\scriptsize max}}
    \end{align*}
    Hint: Expand $\u$ in terms of a basis of orthonormal eigenvectors
    of $A$.
  \end{itemize}
\hfill(10)

%\newpage

\item Compute the complex Fourier series of $\displaystyle f(x)=\cos \left(
    \frac x 2 \right) $ on the interval $[-\pi,\pi]$.
\hfill(10)

%\newpage

\item Recall that the convolution of two functions $\phi$ and $\theta$
  is defined
\[
  (\phi * \theta)(t) = \int_{-\infty}^\infty \phi(s) \, \theta(t-s)
  \, \d s \,.
\]
 Show that
  \[
  \mathcal{F}(f \cdot g) = \frac 1 {\sqrt{2\pi}} \, \tilde
  f * \tilde g \, .
  \]
\hfill(10)

%\newpage

\item
  \begin{itemize}

  \item[(E)] A die is tossed.  Let the random variable $X$ denote the
  value shown.  Define a second random variable by $Y=
  (X-\tfrac72)^2$.  Give the probability functions of both $X$ and
  $Y$. 
  
    \hfill (8)
    
    \item[(A)] The die is tossed a large number of times $N$, each of
    the trials being denoted by a random variable $X_i$.  The central
    limit theorem states that probability function of the random
    variable
    \[
      X=\frac{X_1+\dots+X_N}N
    \] asymptotes to a normal
    distribution with mean $\mu=\tfrac72$ and variance
    $\sigma^2=\sigma_i^2/N$.

    Compute the probability distribution function of
    $Y=(X-\tfrac72)^2$ in terms of $\mu$ and $\sigma$.
      
    \hfill (10)
  \end{itemize}

%\newpage

\item A chip manufacturer produces CPUs that are so complex that it is
  impossible to test every single function of the CPU in a production
  run, but it is possible to design a test that finds a defect with
  probability $p<1$.  Moreover, $\tfrac9{10}$ of all CPUs that are
  produced are defective.  What value of $p$ is needed to guarantee
  that 99\% of all CPUs shipped work?

\hfill (10)


%\newpage

\item What is the probability of drawing exactly $3$ red balls in $4$
  trials from a bag containing $4$ red balls and $2$ green ones, if
\begin{enumerate}
\item a ball drawn is replaced after each trial?
\item a ball once drawn is placed aside?
  \end{enumerate}
\hfill (5+5) 


%\newpage

\item
  \begin{itemize}
  \item[(E)] Show that if $X$ and $Y$ are independent random
  variables, then \[
  \operatorname{Var}[X+Y] = \operatorname{Var}[X] +
  \operatorname{Var}[Y] \,.
\]
  \hfill (8)

\item[(A)] Use the moment generating function
  $M_X(t)=E[e^{tX}]$ to derive the mean and variance of the binomial
  distribution where
\[
  X = \text{number of successes in $n$ trials} \,,
\]
and the probability of success is $p$.
\hfill (10)
  \end{itemize}
 
\end{enumerate}

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