ABSTRACT
In many applications of stochastic processes, the random variable can be a continuous function of the time t. For example, in a population, births and deaths can occur at any time, and any random variable representing such a probability model must take account of this dependence on time. Other examples include the arrival of telephone calls at an office, or the emission of radioactive particles recorded on a Geiger counter. Interpreting the term population in the broad sense (not simply humans and animals, but particles, telephone calls, etc., depending on the context), we might be interested typically in the probability that the population size is, say, n at time t. We shall represent this probability usually by p n ( t ) $ p_{n}(t) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315156576/49190045-e11a-4825-88bf-d327626f56ef/content/inline-math5_1.tif"/> . For the Geiger 1 counter application it will represent the probability that n particles have been recorded up to time t, whilst for the arrival of telephone calls it could represent the number of calls logged up to time t. These are examples of a process with discrete states but observed over continuous times.
