# Abstract:

The paper is concerned with the problem of obtaining the joint distribution of Xt and integrals of the form Yt the integral from 0 to t fXtau, taud tau, where Xt,t or 0 is a continuous time parameter stochastic process appropriately defined on a probability space, with Z as its state space f is a nonnegative measurable function defined on Z x the interval 0 or x infinity. It is assumed that the integral Yt exists and is finite almost surely for every t 0. In another paper, a method was introduced by the author for obtaining the joint distribution of Xt and Yt. This method is based on an auxiliary process Zt called Quantal Response Process, defined as in the text. In this paper, the method is applied to birth processes, both time homogeneous and time nonhomogeneous. The results so obtained are then specialized to case of Simple Epidemic and to certain well-known processes such as Poisson process, Polya process. The method is also applied to certain well-known birth and death processes such as Linear Birth and Death processes with Immigration, and MMI Queue. The paper ends with an application to Illness and Death processes. In each of these cases, distributions of certain useful integrals are explicitly derived by using the above technique. Author

# Subject Categories:

• Statistics and Probability
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