The Poisson Random Variable is a useful model for counts of events occuring within a given time interval, or within a given area, where each event occurs randomly with low probability, and the events are independent of each other.
Many experimental situations occur in which we observe the counts of events
within a set unit of time, area, volume, length etc. For example,
- . The number of cases of a disease in different towns
- . The number of mutations in set sized regions of a chromosome
- . The number of dolphin pod sightings along a flight path through a region
- . The number of particles emitted by a radioactive source in a given time
- . The number of births per hour during a given day
Assumptions: When is the Poisson distribution an appropriate model?[
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The Poisson distribution is an appropriate model if the following assumptions are true.
- K is the number of times an event occurs in an interval and K can take values 0, 1, 2, …
- The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.
- The rate at which events occur is constant. The rate cannot be higher in some intervals and lower in other intervals.
- Two events cannot occur at exactly the same instant.
- The probability of an event in an interval is proportional to the length of the interval.
If these conditions are true, then K is a Poisson random variable, and the distribution of K is a Poisson distribution.