# 12 POISSON POINT RANDOM PROCESSES

## 12.1 INTRODUCTION

Many phenomena are based, in part, on the random arrival of events, and when such events are independent of one another, and occur at a uniform rate, a Poisson point model is appropriate. One important example is that of shot noise. An important property of a Poisson point random process with a constant rate is that it is memoryless. Effectively, it is possible to start, stop, and then restart such a process without altering its statistics after allowing for the stopped time. This property is implicit in the modelling of birth–death random processes. One generalization of a Poisson point process is the Lévy random process. The theory of point random processes is vast and Dayley and Vere-Jones (2003) is a useful reference.

Using, in part, the results from order statistics established in Chapter 11, important results for Poisson point random processes are established for both the finite and infinite intervals. These results include the probability mass function for the number of points in a set interval, the joint probability mass function for the number of points in two disjoint intervals, the probability density function for the *i*th arrival time and *i*th interarrival time, and the joint probability density function for the *i*th and *j*th arrival times and the *i*th and *j*th interarrival times.

### 12.1.1 BACKGROUND RESULTS

The following random variables and associated results are fundamental to the discussion of the Poisson point ...

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