CALCULATION OF EXPECTED VALUES OF OBSERVABLES IN SOME PROBLEMS OF STATISTICAL GEOMORPHOLOGY

In statistical hydrology, three different methods have been used to calculate the expected values of observable quantities in an ensemble. The first is to use an observable as a parameter to characterize the “states” of the ensemble and then to calculate the average value of the observable over all possible states; this is the expected value. The second method is to use an observable as a parameter to characterize the “states” and then to designate the most probable value of the observable as the expected value; this is known as the “minimum variance principle”. Finally, the third method is to use a parameter different from an observable to characterize the “states” and then to take the value of the observable for the most probable state as the expected value. Of all these different methods, only the first one is, in principle, physically correct. Thus, a study was made to determine the situations where methods other than the first can be used to calculate the expectation values of observables. It was found that this can be done only in those situations where all of the following conditions hold: (i) The number of parameters used to characterize the states is finite; (ii) these parameters are either the same as the observables or are explicitly related to the observables; and (iii) each of the parameters has a symmetrical and unimodal probability distribution. The cases where the minimum variance principle is valid, are thereby delineated.