A unified study of the multivariate joint probability function of the state variables with quantized levels for a stochastic environmental system with discrete data: Theory and experiment

Abstract

All the information on statistical properties of the state variables of a stochastic system can be derived by first finding the multivariate joint probability function of their variables. Furthermore, random data are often obtained in digital form at discrete time intervals. Accordingly, theoretical consideration is given, first, to the joint probability function with quantized levels and its joint factorial moments, which are suited to an actual situation where real experimental data are taken in quantized form: i.e., as finite sets of discrete numbers. As a special case when the level width tends to 0, the above theory includes the well-known expansion series distribution in continuous form. Second, the validity of the theory is confirmed both by means of digital simulation and by application to experimentally observed road traffic noise and environmental noise data.