3.6.3 The Poisson Process

Application of the Poisson distribution:

• Occurrence of events over time
  • Examples of events:
    • Visits to a particular website
    • Pulses recorded by a counter
    • Email messages sent to a specific address
    • Accidents in an industrial facility
    • Cosmic ray showers observed at an observatory

Assumptions about event occurrence:

1. Parameter $\alpha >0$:
   • For any short time interval of length $\Delta t$:
   • Probability of exactly one event:
     • $\alpha \cdot \Delta t+o(\Delta t)$
2. Probability of more than one event during $\Delta t$:
   • $o(\Delta t)$
   • Implies probability of no events during $\Delta t$:
     • $1-\alpha \cdot \Delta t-o(\Delta t)$
3. Independence:
   • Number of events during $\Delta t$ is independent of the number prior to this interval.

• Informal interpretation of Assumption 1:
  • Probability of a single event occurring in a short time interval is:
  • Approximately proportional to the length of the interval
  • $\alpha$ is the constant of proportionality
• Notation:
  • Let $P_k(t)$ denote Probability that $k$ events will be observed during a time interval of length $t$

Proposition

The number of events during a time interval of length $t$ is a Poisson rv with parameter $\mu =\alpha t$ $P_k\left(t\right)=e^{-\alpha t}\cdot {\left(\alpha t\right)}^k/k!.$

The expected number of events during any such time interval is then $\alpha t$ , so the expected number during a unit interval of time is $\alpha$ .

• The occurrence of events over time as described is called a Poisson process;
• the parameter $\alpha$ specifies the rate for the process.
• EX 3.41 pulse
• Observing events in space:
  • Context:
    • Instead of observing over time, consider a two- or three-dimensional region
  • Example:
    • Select a region $R$ on a map of a forest
    • Count the number of trees
    • Each tree represents an event at a particular point in space
• Assumptions:
  • Similar to assumptions 1-3 for events over time
• Result:
  • Number of events in region $R$:
  • Follows a Poisson distribution
  • Parameter: $\alpha \cdot a(R)$
    • $a(R)$ is the area of region $R$
    • $\alpha$ is the expected number of events per unit area or volume