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Give paragraph-leading quotations more prominence in H. Math section
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Michael Norrish committed Jun 12, 2020
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24 changes: 13 additions & 11 deletions Manual/Description/math.stex
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Expand Up @@ -795,9 +795,10 @@ probability space with its target measurable space:

\paragraph{Independence}

``The concept of mutual independence of two or more experiments holds,
\begin{quote}\textit{The concept of mutual independence of two or more experiments holds,
in a certain sense, a central position in the theory of
probability.''~\cite[p.~8]{Kolmogorov:1950}
probability.}\\\hfill{}Kolmogorov~\cite[p.~8]{Kolmogorov:1950}
\end{quote}
Informally, events or random variables are independent if they do not
affect each other's probabilities. Thus, two events $A$ and $B$ are independent
if $\mathscr{P}\{A \cap B\} = \mathscr{P}\{A\}\mathscr{P}\{B\}$. Moreover, a finite
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\end{hol}

\paragraph{Mathematical Expectation}

``To achieve reasonable simplicity it is often necessary to describe
\begin{quote}\textit{
To achieve reasonable simplicity it is often necessary to describe
probability distributions rather summarily by a few `typical
values.' ... Among the typical values the \emph{(mathematical) expectation},
or \emph{mean}, is by far the most important.
It lends itself best to analytical manipulations, and it is
preferred by statisticians because of a property known as sampling stability.''~\cite[p.~221]{Feller:2004vv}
preferred by statisticians because of a property known as sampling stability.}\hfill{}Feller~\cite[p.~221]{Feller:2004vv}
\end{quote}
Formally, expectation is nothing but a synonym of (Lebesgue) integration:
\begin{hol}
\begin{alltt}
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\paragraph{Limit theorems and the laws of large numbers}

``In the formal construction of a course in the theory of probability, limit theorems appear
as a kind of superstructure over elementary chapters, in which all problems have finite,
purely arithmetical character. In reality, however, the epistemological value of the theory
of probability is revealed only by limit theorems. Moreover, without limit theorems it is
impossible to understand the real content of the primary concept of all our sciences – the
concept of probability.''~\cite[p.~1]{Gnedenko:1954vf}
\begin{quote}
\textit{
In the formal construction of a course in the theory of probability, limit theorems appear as a kind of superstructure over elementary chapters, in which all problems have finite, purely arithmetical character.
In reality, however, the epistemological value of the theory of probability is revealed only by limit theorems.
Moreover, without limit theorems it is impossible to understand the real content of the primary concept of all our sciences---the concept of probability.}\\\hfill\mbox{Gnedenko~\cite[p.~1]{Gnedenko:1954vf}}
\end{quote}

It is sometimes convenient to think of probabilities intuitively as
limits of observable \emph{frequencies} in repeated experiments. This
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