Understanding Poisson Distribution: Explanation & Examples

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In summary, the Poisson distribution, as explained in the reference book, is a random experiment called a "Poisson process" where events occur at random throughout an interval of real numbers. If the interval can be divided into smaller subintervals where the probability of more than 1 event is 0, the probability of 1 event is proportional to the length of the subinterval, and the events in each subinterval are independent, then it is considered a Poisson process. This means that in small intervals, the probability of multiple events decreases quickly while the probability of 1 event remains proportional to the length of the interval. However, the probability is never 0 as long as the parameter x is greater than 0.
  • #1
nothGing
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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..
 
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  • #2
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
 
  • #3
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.
 
  • #4
[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..
 
  • #5
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.
 

1. What is Poisson distribution?

Poisson distribution is a probability distribution that is used to model the number of times an event occurs within a specific time or space. It is also known as a discrete probability distribution, as it can only take on integer values.

2. How is Poisson distribution different from other probability distributions?

Poisson distribution differs from other probability distributions in that it is used to model the number of occurrences of an event within a specific time or space, rather than the probability of a certain outcome. It also assumes that the events occur independently and at a constant rate.

3. What type of data is suitable for Poisson distribution?

Poisson distribution is suitable for count data, which is data that represents the number of occurrences of a particular event. For example, the number of car accidents in a day, the number of customers who enter a store in an hour, or the number of calls received by a call center in a day.

4. How is Poisson distribution calculated?

Poisson distribution is calculated using the formula P(x) = (e^-λ * λ^x) / x!, where x is the number of occurrences of the event, and λ is the average number of events that occur in the given time or space.

5. What are some real-life examples of Poisson distribution?

Poisson distribution can be seen in a variety of real-life scenarios, such as the number of patients arriving at a hospital emergency room in a given hour, the number of defects in a batch of products, or the number of earthquakes in a specific region in a year.

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