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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Review for Final Exam}
\date{Thursday, May 23, 12:30-14:30 in the Campus Center East Wing and Conference 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 problems:} See, in particular, Question~6 on Midterm~1 from
Spring 2003, Question~6 on Midterm~1 from Spring 2004, and Homework~4
Question~2.

\item Inner products: Definition, examples, orthonormal basis,
projections.

\item Gram-Schmidt orthonormalization.

\emph{Study problem:} Midterm 2 Retake, Question~2.

\item Can you show that the eigenvectors of a symmetric (Hermitian)
matrix are orthogonal provided the eigenvalues are distinct?

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

\emph{Study problem:} Midterm 1 Question~5A (some versions of the exam
have a different order of questions -- this reference is uses the order
in the file \texttt{midterm1-solutions.pdf}).

\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.

\item Delta function: basic properties and simple computations.

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

\item Bayes' rule.

\emph{Study problems:} Think through the Monty Hall problem and its
formalization via Bayes' rule again; Homework~8 Question~6, and the
following problem (which is slightly larger than the problems which
will be on the exam, but gives a rather typical Bayes' rule setting):

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, probability distribution
functions for continuous random variables, sums of identically and
independently distributed random variables (``the road to the central
limit theorem'').

\emph{Some 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}



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