Understanding Poisson Distribution: Explanation & Examples

[nothGing]

The explanation for the Poisson distribution in reference book is "
when given an interval of real number, assume events occur at random throughout the interval. If the interval can be partitioned into subintervals of small enough length such that
1. the probability of more than 1 event in a subinterval is 0
2. thw probability of one events in a subinterval is the same for all subintervals and proportional to the length of the subinterval, and
3. the event in each interval is indepedent of other subintervals, the random experiment is called " POISSON process". "

But i don't really understand what is it mean for part 1.
Can anyone explain to me?
thx..

 

[statdad]

the probability of more than 1 event in a subinterval is 0

Suppose the interval is time: this means that during a short enough time interval the chance of having multiple occurrences of the event is zero.

Suppose the "interval" is a region of area (you are looking at paint flaws in a newly manufactured car, as an example): if you look at a small enough area the chance of having multiple flaws is 0

 

[mathman]

1. the probability of more than 1 event in a subinterval is 0

This is misleading, since the probability is never 0, although it can be vanishingly small compared to the probability of 1 event. For small intervals, the ratio is proportional to the length of the interval.

 

[nothGing]

[This is misleading, since the probability is never 0, although it can be vanishingly small compared to the probability of 1 event. ]
"Mathman", I don't really understand what do you mean since it's different way of explanation from "statdad".
Can you explain some more? thx..

 

[mathman]

P(n events in an interval) is e^-x xⁿ/n!, where x is some parameter.
For intervals, x is proportional to the length of the interval. P(n=2)/P(n=1) = x/2, while P for larger n disappear more quickly.
However no matter how small the interval is, the probability is not 0, as long as x > 0.