Mean and Variance

A random function $Z=Z(/omega,/boma{x})$ depends on two arguments, one is for the elementary random events $/omega$ and the other one for the geographical place $/boma{x}$. In this subsection we only consider a particular place $/boma{x}$, such that $Z(/boma{x})$ may be seen as a usual random variable. An important tool for the description of the distribution is the notion of the (mathematical) expectation $EZ$ of the random variable $Z$. It is defined by the weighted mean of all possible values, weighted by the probability density $f$, namely

The integration is over the whole domain of values, e.g. in the case of grades in % it would be from 0 to 100%. In case of only finite many possible values $c_{j}$ with corresponding probabilities $f_{j}, j=1,/ldots,k,$ we simply use the sum:

This theoretical expectation of a random variable is often simply called mean, $m=EZ$.

The variance $/sigma^{2}$ of a random variable is defined by the expected value (i.e. the mean) of the quadratic deviation from the mean $m$. In the general case we have

and in the case of finite many possible values

The square root of the variance is also called standard deviation or (sometimes) dispersion. It represents a measure for the width of the distribution, i.e. for the variability of $Z$. The unit of the dispersion is of course the same as that of the variable $Z$, and we may define e.g. a tolerance interval $m /pm 2 /sigma,$ which says that a realization will fall in this interval with a certain (high) probability.

In practice, if $n$ empirical measurements (realizations) $z_{1},/ldots,z_{n}$ of a random variable are at hand, the expectation may be estimated by the simple calculation of the mean. For this estimated mean $/hat{m}$ we have

and for the estimated variance

where the denominator in general is $n-1$ to compensate the loss of information because of estimation of $m$.