

\subsection{\bf Means---Moments, etc.}
\subsection{\bf for One-dimensional Distributions}
\begin{slide}
\begin{description}
\item{$\bullet$} Given a one-dimensional distribution, we can define:
\end{description}
\begin{itemize}
\item {\em mean:}\ \ $\langle  x\rangle = 
{\displaystyle \int\limits^{+\infty}_{-\infty}} xp(x) dx$
\item {\em n'th moment:}\ \ $\left\langle x^n \right\rangle = {\displaystyle
\int\limits^\infty_{-\infty}} x^n p(x)$ dx
\item {\em expectation:}\ \ $\langle f(x)\rangle = {\displaystyle
\int\limits^\infty_{-\infty}} f(x) p(x) dx$ which ``follows'' from {\em
n}'th moment by Taylor expansion of $f(x)$.  This can be considered as
the mean of the random variable $z=f(x)$.

\item {\em n'th central moment:}\ \ $\left\langle (x-\langle x\rangle
)^n \right\rangle = {\displaystyle \int\limits^{+\infty}_{-\infty}}
(x-\langle x\rangle)^n \, p(x) dx$

\item {\em Variance:}\ \ $V_x = \left\langle (x-\langle x\rangle )^2
\right\rangle =$ 2nd central moment

\item {\em Standard deviation $\sigma_x$:}\ \ $V_x = \sigma^2_x$ where
$\sigma_x$ is used much more than $V_x$.

\item Note:
\begin{equation}
V_x = \left\langle x^2 \right\rangle - 2\langle x\rangle \langle x\rangle
+ \langle x\rangle^2 = \left\langle x^2\right\rangle - \langle x\rangle^2
\end{equation}
a result which is used continuously.
\end{itemize}
\end{slide}

