varianceVariance2001-10-26 02:33:412007-07-08 11:26:35Definition% The standard font packages
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\subsection*{Definition}
The \emph{variance} of a real-valued random variable $X$ is
\[
\Var X = \E\bigl[ (X - m)^2 \bigr]\,, \quad m = \E X\,,
\]
provided that both expectations $\E X$ and $\E[(X-m)^2]$ exist.
The variance of $X$ is often denoted by $\sigma^2(X)$, $\sigma^2_X$,
or simply $\sigma^2$.
The exponent on $\sigma$ is put there so that the number
$\sigma = \sqrt{\sigma^2}$
is measured in the same units as the random variable $X$
itself.
The quantity $\sigma = \sqrt{\Var X}$ is called the \emph{standard deviation}
of $X$;
because of the compatibility of the measuring units,
standard deviation is usually the quantity that is quoted
to describe an emprical probability distribution, rather than the variance.
\subsection*{Usage}
The variance is a measure of the dispersion or variation
of a random variable
about its mean $m$.
It is not always the best measure of dispersion for all random variables,
but compared to other measures,
such as the absolute mean deviation, $\E[ \abs{X-m} ]$,
the variance is the most tractable analytically.
The variance is closely related to the $\Le^2$ norm for
random variables over a probability space.
\subsection*{Properties}
\begin{enumerate}
\item
The variance of $X$ is the second moment of $X$ minus
the square of the first moment:
\[
\Var X = \E[X^2] - \E[X]^2\,.
\]
This formula is often used to calculate variance analytically.
\item
Variance is not a linear function. It scales quadratically,
and is not affected by shifts in the mean of the distribution:
\[
\Var[ aX + b ] = a^2 \Var X\,, \quad \text{ for any $a, b \in \real$.}
\]
\item
A random variable $X$ is constant almost surely if and only
if $\Var X = 0$.
\item
The variance can also be characterized as
the minimum of expected squared deviation of a random variable from any point:
\[
\Var X = \inf_{a \in \real} \E[(X-a)^2]\,.
\]
\item
For any two random variables $X$ and $Y$ whose variances exist,
the variance of the linear combination $aX + bY$
can be expressed in terms of their covariance:
\[
\Var[aX+bY] = a^2 \Var X + b^2 \Var Y + 2ab \Cov[X,Y]\,,
\]
where $\Cov[X,Y] = \E[(X-\E X)(Y-\E Y)]$,
and $a, b \in \real$.
\item
For a random variable $X$, with actual observations $x_1, \dotsc, x_n$,
its variance is often estimated
empirically with the \emph{sample variance}:
\[
\Var X \approx s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2\,,
\quad
\bar{x} = \frac{1}{n} \sum_{j=1}^n x_j\,.
\]
\end{enumerate}