Some Properties of Distributions Useful in the Study of Rare Events

This paper presents techniques that facilitate estimation of the parameters of a gamma or beta distribution when two percentiles or one percentile and the mean value are known. Similarly, when the values of the parameters are known, simple techniques are developed to estimate the percentiles. These techniques apply when the random variable is small, e.g., when these distributions are the state-of-knowledge distributions of failure rates (failures per hour or demand). A further result of the paper is that, for distributions with one shape parameter, the mean value possesses a minimum, when an upper percentile is held constant. Expressions for these minima are derived for several useful distributions.