Random function first-order moment

In practice, only the first two moments of the random function are of interest. The first order moment is the expectation (mean) of the random function at an arbitrary location x, which is defined to be... 

To describe single-point measurements of a random process, we use the first-order probability density function p/(/). Then p/(/) df is the probability that a measurement will return a result between / and / -I- df. We can characterize a random process by its moments. The nth moment is the ensemble average of /", denoted (/"). For example, the mean is given by the first moment of the probability density function. 

The random function is said to be weakly or second order stationary when its first two moments are invariant under simultaneous translation by h. That is, for every x and h ... 

Thus, if the assumption of second order stationarity holds, then statistical inferences about the first two moments become possible since each pair of observations that are separated by a distance h can be considered a different realization of the random function. 

The present study shows that It is possible to evaluate the variability of statically determinate and statically indeterminate structures due to spatial variation of elastic properties without resort to finite element analysis. If a Green s function formulation is used, the mean square statistics of the indeterminate forces are obtained in a simple Integral form which is evaluated by numerical methods in negligible computer time. It was shown that the response variability problem becomes a problem Involving only few random variables, even if the material property is considered to constitute stochastic fields. The response variability was estimated using two methods, the First-Order Second Moment method, and the Monte Carlo simulation technique. 

Just as a random variable is characterized by the moments of its distribution, a stochastic process is characterized by its time correlation functions of various orders. In general, there are an infinite number of such functions, however we have seen that for the important class of Gaussian processes the first moments and the two-time coiTelation functions, simply referred to as time correlation functions, fully characterize the process. Another way to characterize a stationary stochastic process is by its spectral properties. This is the subject of this section. 

---