Poisson

What It is

The Poisson distribution characterizes the number of events occurring within a given time interval. It has a single parameter that results in many shapes based on its magnitude:

λ rate of occurrences

Classically, this distribution is used to describe the number of arrivals into a system. For example, assume the above plot is of 1000 evaluated time intervals wherein the number of events that occurred is recorded. The value of λ = 25 indicates that within the time interval, an average of 25 events occur. In some intervals, more/less than 25 events occur, with decreasing likelihood. Notice that it is very unlikely for only ten or 44 events to occur.

Variants

The single parameter alters the centrality of the distribution as well as the range. The following plots illustrate the the shift and the increase in range as the value of λ increases.

λ = 1

Expecting on average only one event per time interval implies that there will be occurrences of more and less number of events. Specifically, there is a large number of occurrences of zero events happening.

λ = 35

Increasing the expected number of events shifts the distribution. The most likely outcomes are centered around the value of λ. The spread of the distribution also increases. Note this difference when comparing this plot to that where λ = 1. (more shapes)

In Real Life (IRL)

To apply the Poisson distribution IRL, you must be sure that the occurrence of the events of interest are independent. That is, the happening of one event can not effect successive events, or be effected by previous events.

Example A

You have now been appointed as President of Special Projects. You envision a retail space to expand your currently online-only business. Extensive research on customer bases and market share is completed. Finance believes the minimum number of visits by customers required to ensure profitability be 25 customers per ten hour day of operation. The data scientist, using data on neighboring stores, believe that on average there are 3 visits per hour. With these predictions, will the store be survive?

You realize that there is not much math to do here. You define the time interval to be one day, define λ = 3 * 10 = 30, and plot it.

A majority of the outcomes exceed the threshold. It appears as though your store will be successful.

Example B

Now, as a business process expert, a business associate desires to characterize new work requests he receives. The associate wants to consider organic requests as well as referrals, knowing that about 50% of work comes from them. Thus, it is believed the number of requests can be modeled as arrivals. Historically, your associate receives an average of twelve requests for work. A Poisson distribution is plotted with λ = 12 to understand a potential range of number of requests.

The associate deducts from the plot that practically, between about five and twenty requests could be made per month.

Upon review, you notice that the referrals are dependent occurrences, not independent of the original referring request for work. You Inform the associate that it would be safer to extract the referrals when determining λ. Thus, you define λ = 6 and plot it.

You deduce that the expected number of organic requests would range between one and thirteen requests.

You advise that there needs to be another process for analyzing the referrals.

It may be possible that 50% of the average number of requests refer a new customer. The resulting range would then be between:

low 0 + 6 * 0.5 = 3

high 14 + 6 * 0.5 = 17

Conclusion

The Poisson distribution can be applied to any system where events do not effect each other, given a consistent time interval to evaluate. That is why it has classically been used in simulation and queuing theory to model arrivals to a system. Here, we have shown how it gives us a better understanding about the ranges and likelihoods of number of events occurring.

YHWH, we declare that through you all good things are possible. We thank you for each new and different outcome each day.

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