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\newpage
\section{Probability}
\label{sec:prb}

See \cite{bib:griffiths}, \cite{bib:kerl}, \cite{bib:gs}, and \cite{bib:fg},
respectively, for treatments of increasing levels of rigor.

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\subsection{Probability spaces and probability measures}

$(\Omega, \mcF, P)$ from prb MT appx.  Meas. space, prob. space;
$\mcF=\mathrm{Bo}(\R)$.

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\subsection{Random variables}

fcn from prb spc to meas spc.  make sure to note special case $\Omega=\R^d$ and
$X$ is identity function.  My convention about ``random variable'' in this
document.

$P(X\in[a,b])=P(X^{-1}([a,b]))$.

$P(X\in A)=P(X^{-1}(A))$.

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\subsection{Distribution functions, mass functions, and density functions}

CDFs $F_X(x)$.

PDFs when they exist:  $f_X(x)$.  CRVs.

PMFs when they exist:  $f_X(x)$.  DRVs.  (Need both for QM.)

$P(X \in A) = \int_A f_X(x)\,dx$ for all Borel $A$.

\beginex
\label{ex:Cauchy_PDF}
The \bfidx{Cauchy random variable} $X$ has PDF
$$
	f_X(x) = \frac{1}{\pi} \; \frac{1}{x^2+1}.
$$
You can check using freshman-calculus techniques that this has
integral 1.
\mendex

\beginex
\label{ex:standard_normal_PDF}
The \bfidx{standard normal random variable} $X$ has PDF
$$
	f_X(x) = \frac{1}{\sqrt{2\pi}} \; e^{-x^2/2}.
$$
This also has integral 1.
\mendex

What is the relationship between discrete/continuous CDFs and
discrete/continuous spectra?  Spell it out here or in a separate appendix.  Is
there any relationship at all?  What about empty spectrum?  What is ``residual
spectrum''?  Wiki articles on decomposition of spectrum, Borel functional
calculus,
Lebesgue's decomposition theorem,
Stone's theorem on one-parameter unitary groups.

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\subsection{Expectations}

$E[X]$ or $\langle X\rangle$: $\sum x f_X(x)$ or $\int x f_X(x)\,dx$.
Absolute-integrability requirement.

$\Var(X) = E[X^2]-E[X]^2$.  $\sigma_X(x)=\sqrt{\Var(X)}$.

\beginex
\label{ex:Cauchy_mean}
The Cauchy random variable from example \ref{ex:Cauchy_PDF} has infinite mean:
use $u$-substitution on the parts $x\ge 0$ and $x \le 0$ and you will find that
the mean is undefined because $x\,f_X(x)$ is not absolutely integrable.  The
problem is that $f_X(x)$ goes to zero at infinity but not fast enough.
\mendex

\beginex
\label{ex:standard_normal_mean}
The standard normal random variable from example \ref{ex:Cauchy_PDF} has mean
0, as you can verify using $u$-substitution.  Unlike the Cauchy random
variable's PDF, this PDF goes to zero at infinity at a higher rate.
\mendex

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\subsection{Independence}

IID

%% ----------------------------------------------------------------
\subsection{The IID paradigm}

Let $X$ be a sequence of IID RVs.  Form $\Xnbar=\sum X_i/n$.  (This is called
the \bfidx{sample mean}.) Cite the LLN.  Note that
$\langle\Xnbar\rangle=\langle X\rangle$.  Also $\Var(\Xnbar)=\sigma^2/n$, so
$\sigma(\Xnbar)=\sigma/\sqrt{n}$.  Emphasize that this is the\index{standard
deviation of the mean} standard deviation \emph{of the mean itself}.

Example:  Experiment is 1000 tosses of a fair coin.  (Or, toss 1000 fair coins
all at once).  Then count heads.  1st time:  Maybe 510.  2nd time: 493.  3rd
time: 507.  Etc.  Do \emph{that} many times and the average of your results
will tend toward 500.  Compute the sample standard deviations of your data; the
result will tend toward will be $\sqrt{4000}/\sqrt{N}$.  Elaborate.

Caveat about wave-function collapse for QM --- do \emph{identical} experiments,
not \emph{repeated} experiment.




basic facts about probability functions, with refs

expectations, mean, median \ldots

variance and stddev

multiple RVs:  mention joint PDF.

maybe mention LLN and CLT

PRB, FA/LA, PDEs/ODEs.

Cplx PDF: $\langle Q \rangle = \sum/\int \phi^* Q \phi$

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\subsection{Multiple random variables}

joint CDF and joint PDF.

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