8. PROBABILISTIC APPROACH FOR UNCERTAINTY ANALYSIS

Probabilistic approach is the most widely used technique for uncertainty analysis of mathematical models. There are a number of text books that describe the concepts and application of probabilistic analysis in detail. Tsokos [201] presents excellent introductory material for probabilistic analysis, Johnson [114] explains the multivariate random variables, and Papoulis [160] presents an excellent description on probability and random variables from a mathematical view point. Additionally, Gardiner [75] presents the applications of probabilistic analysis in modeling. This appendix attempts to merely summarize some basic information on probability and random variables, which can be found in more detail in the abovementioned texts.

In the probabilistic approach, uncertainties are characterized by the probabilities associated with events. An event corresponds to any of the possible states a physical system can assume, or any of the possible predictions of a model describing the system. In the study of environmental pollution, the situation where the contaminant concentration exceeds a regulatory level can be an event. Similarly, in case of mathematical modeling, the situation where the model outputs fall in a certain range, is also an event. In short, any possible outcome of a given problem, such as the tossing of a coin, experimental measurement of contaminant concentration,as and mathematical modeling of a physical system, is an event. The probability of an event can be interpreted in terms of the frequency of occurrence of that event. When a large number of samples or experiments are considered, the probability of an event is defined as the ratio of the number of times the event occurs to the total number of samples or experiments. A probability of 0 for an event means that the event will never occur, and a probability of 1 indicates that the event will always occur. Examples of experiments or samples include repeated coin tossing, a large number of independent measurements of contaminant concentration, or a large number of simulations using a mathematical model with randomly varying parameters.

The following examples illustrate the concept of probability based on a large number of samples: (a) if a coin is tossed, the probability of heads turning up is 0.5 and that of the tails is 0.5. This means that if the coin is tossed a large number of times, heads will turn up roughly half the time and tails half the time. (b) the statement that the probability that a pollutant concentration $c$ lies between $c_1$ and $c_2$ equals $p$ means the following: from a large number of independent measurements of the concentration $c$, under identical conditions, the number of times the value of $c$ lies between $c_1$ and $c_2$ is roughly equal to the fraction $p$ of the total number of samples.