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We will consider a process in which points occur randomly in time. The phrase points in time is generic and could represent, for example:
It turns out that under some basic assumptions that deal with independence and uniformity in time, a single, one-parameter probability model governs all such random processes. This is an amazing result, and because of it, the Poisson process (named after Simeon Poisson) is one of the most important in probability theory.
There are two collections of random variables that can be used to describe the process; these collections are inverses of one another in a certain sense.
First, let denote the time of the arrival for . Next, let denote the number of arrivals in the interval for . Note that
since each of these events means that there are at least arrivals in the interval .
The assumption that we will make can be described intuitively (but imprecisely) as follows: If we fix a time , whether constant or one of the arrival times, then the process after time is independent of the process before time and behaves probabilistically just like the original process. Thus, the random process has a strong renewal property. Making the strong renewal assumption precise will enable use to completely specify the probabilistic behavior of the process, up to a single, positive parameter.
Think about the strong renewal assumption for each of the specific applications given above.
As a first step, note that the strong renewal assumption means that the times between arrivals, known as interarrival times, must be independent, identically distributed random variables. Formally, the sequence of interarrival times is defined as follows:
The strong renewal assumption will allow us to find the distribution and essential properties of each of the following in turn:
The Poisson process is the most important example of a type of random process known as a renewal process. For such processes generally, the renewal property must only be satisfied at the arrival times; thus, the interarrival times are independent and identically distributed. A separate chapter on Renewal Processes explores the processes in detail.