Improving the Variability Function in Case of a Uni-Modal Probability Distribution

Abstract

For predicting a future development in a reliable and precise way, one must necessarily develop a stochastic model given by a Bernoulli-Space introduced by von Collani in 2004. The core of the stochastic model is a set of probability distributions, each describing one potential future development. Any probability distribution for a real-world phenomenon is uniquely determined by the range of variability and the values of a certain number of moments of the considered random variable. Thus, the problems arise how to determine the range of variability on the one hand and the values of the necessary moments on the other. While there are a number of statistical methods to deal with the later problem, the former one is hardly mentioned in any statistical textbook. This is a surprising fact as the range of variability of a random variable is most important and in the special case of a uniform distribution it even determines uniquely the probability distribution and, hence, the values of all moments. Besides the principle importance of the bounds of the range of variability, they are often of great significance, if the extreme values of a random variable are of interest, e.g. for natural phenomena as floods or weather conditions or for technical threshold values. Moreover, the extreme values of a relevant random variable are in general of special interest in the design, maintenance and quality control of systems and facilities. The reason, why there are almost no statistical methods available for determining the range of variability of a given random variable, is the frequently made unrealistic assumption that the range of variability is unbounded. This paper deals with developing a method for improving the knowledge about the range of variability of a given random variable based on the Bernoulli Space as stochastic model of uncertainty. Stübner et al. (2004) derived a general method for improving the variability function χ of a given Bernoulli Space . Subsequently, this method has been applied to the case of uniform distributions by Sans et al. (Economic Quality Control 20: 121-142, 2005) and to the case of monotonic probability distributions by Sans et al. (Economic Quality Control 20: 121-142, 2005). In this paper the method is extended to the important case of the family of uni-modal probability distributions.