A Physical Basis for the Generalized Gamma Distribution

Introduction. A number of known families of probability distributions can be derived from requirements that are physical in the sense that they describe the random behavior of the event under consideration. The Poisson process is typical of these: If X{t) is equal to the number of occurrences of a specified event in the interval [0, t), then one can show that X (t) has, for each t, a Poisson distribution, if the probabilistic behavior of the event satisfies a few very simple physical requirements [1], The normal distribution may also be derived from a few physical requirements. In this note we shall employ a simple model and statistical-mechanical methods to derive the three-parameter generalized gamma distribution. The history of this family of distributions was reviewed and further properties were discussed in 1962 by Stacy [2], Subsequent work on statistical problems associated with the distribution has been done by Bain and Weeks [3]. Special cases of the generalized gamma distribution include the Weibull, gamma, Rayleigh, exponential and Maxwell velocity distributions. Another special case is a distribution recently derived on a statistical-mechanical basis to describe rainfall run-off from a watershed [4], The model. We shall consider the following situation: The occurrence of an event, such as the failure of a component or system, depends on some variable such as the stress to which the part has been subjected or the time during which it has been subjected to a given level of stress or use. This variable (be it stress or time) will be designated by t and the number of occurrences of the event during the interval , t,) will be designated by N{ , where t, — i,_i = At and t0 is the arbitrary origin.** The requirements that we shall impose upon the iV.'s are as follows: 1. The total number of occurrences of the event is fixed: