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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Review for the Final Exam}
\date{May 21, 2004, 8:00--10:00, Naber Lecture Hall}
\maketitle

\begin{enumerate}

\item Solve a system of linear equations, matrix inversion.

\emph{Study problems:} There are plenty of practice problems on
homework, past exams, and in the book.

\item Concept of vector space, linear independence, basis; linear
transformations: definition, representation by a matrix; change of
basis.

\emph{Study problem:} See, in particular, question 6 on Midterm 1 from
Spring 2003.

\item Eigenvalues and eigenvectors, in particular for Hermitian
matrices; determinants; diagonalization; inner products.

\emph{Study problems:} See, in particular, questions~1 and~2 of Midterm~2
from Spring 2003, and questions~2 and~6 from Homework~5.

\item Fourier Series: compute the series, properties.  Concentrate on
the complex Fourier series!

\emph{Study problems:} There are plenty of practice problems on
homework, past exams, and in the book.

\item Fourier transform: Compute the transform, inverse Fourier
transform, properties, Fourier transform of the convolution of two
functions, Parseval theorem.

\emph{Study problem:} Recall that the convolution of two functions
$f$ and $g$ is defined by
\[
  (f * g)(x) = \int_{-\infty}^\infty f(y) \, g(x - y) \, \mathrm{d} y
  \,.
\]
Show that $\mathcal{F}(f * g) = \sqrt{2 \pi} \, \tilde f \, \tilde
g$. 

\item Delta function: Definition, representations; application of the
delta function when computing the probability distribution function
for functions of a continuous random variable.

\emph{Study problem:} The velocity of a molecule in a one-dimensional
gas is modeled by a continuous random variable $V$ whose probability
distribution function is normal distribution with mean $0$ and
variance $\sigma^2$,
\[
  f(v) = \frac1{\sqrt{2\pi} \sigma} \, \mathrm{e}^{-\tfrac{v^2}{2
  \sigma^2}}  
  \,. 
\]
The energy of each molecule is the random variable $E=V^2$.  Compute
the probability distribution function for $E$.

(For physicists: $\sigma^2 = kT/m$, where $k$ is the Boltzmann
constant, $m$ the molecular mass, and $T$ the temperature.  In
physical units, we have to write $E=m\,V^2$.  For background
information, see
\texttt{http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/kintem.html}.)

\item Probability: Outcomes, events, sample spaces, definition of
probability, conditional probability, Bayes' rule.

\emph{Study problem:} A machine sorts potatoes into small, medium and
large ones, but it makes mistakes. Suppose that the probability that a
random potato will end up in a specified category is according to the
following table.
\begin{center}
    \begin{tabular}[t]{r|c|c|c|}
      &\multicolumn{3}{c|}{categorized as}\\
      potato of size & small & medium & large \\\hline
      small & 0.8  &   0.15 &   0.05 \\
      medium  &  0.2  &  0.7  &  0.1 \\
      large & 0.05 &   0.1 &   0.85
    \end{tabular}
\end{center}
Suppose further that 30\,\% of the potatoes are small, 45\,\% are
medium and 25\,\% are large. What are the percentages of small, medium
and large potatoes in each of the three piles the machine produces?

\item Permutations and Combinations; their use for the computation of
probabilities; expected value and variance; binomial, Poisson, and
Gaussian distribution; know how to compute the mean and variance using
the moment generating function.

\emph{Study problem:} In a game you throw a pair of dice. If the sum
of the values equals 12 you win 5 euros, if it equals 11 you win 1
euro; for all other outcomes you won't get anything.
\begin{enumerate}
\item Calculate the probability of winning exactly 6 euros after 6
games.
\item Compute the expected value of the pay-out per game.
\end{enumerate}

\item Properties of random variables, functions of random variables.

\emph{Questions to ask:} If $X$ and $Y$ are random variables,
determine if, or under what conditions, the following statements are
true: $E[XY]=E[X]\,E[Y]$, $\Var[XY]=\Var[X] \, \Var[Y]$,
$E[X+Y]=E[X]+E[Y]$, $\Var[X+Y]= \Var[X]+\Var[Y]$?

\end{enumerate}




\noindent Notes:

\begin{itemize}

\item No calculators.

\item Many computations will involve complex numbers.  If you have
difficulties with complex numbers, you should practice manipulating
them.

\item ESM 2A homework sheets for Linear Algebra and Probability are on
the web and a good source for practice problems.

\texttt{http://math.iu-bremen.de/stoll/teaching/ESM2A-2003-Spring/schedule.html}
\item The following topics should also be revised, as they may be
required as part of some question: Equations for lines and planes;
distance of a point to a line or plane; distance between two lines;
Orthonormal bases, Gram-Schmidt orthnormalization, Hermitian Matrices
and operators.


\end{itemize}



\end{document}
